Real Analysis I
Purpose of Course showclose
This course is designed to introduce you to the rigorous examination of the real number system and the foundations of calculus of functions of a single real variable. Analysis lies at the heart of the trinity of higher mathematics – algebra, analysis, and topology – because it is where the other two fields meet.
In calculus, you learned to find limits, and you used these limits to give a rigorous justification for ideas of rate of change and areas under curves. Many of the results that you learned or derived were intuitive – in many cases you could draw a picture of the situation and immediately “see” whether or not the result was true. This intuition, however, can sometimes be misleading.
In the first place, your ability to find limits of realvalued functions on the real line was based on certain properties of the underlying field on which undergraduate calculus is founded: the real numbers. Things may have become slightly more complicated when you began to work in other spaces. For instance, you may remember from multivariable calculus (calculus in three or more real variables) that for some functions there were points where some directional derivatives existed and others did not. In fact, there exist other more exotic spaces where other complications arise.
In the second place, the techniques that you used to find limits may have been very informal. In this course, you will learn to rigorously justify every step in the limiting process or proof. Learning to do this well in the familiar context of the real line, will prepare you for wilder, more complicated mathematical situations. After a brief review of set theory, you will dive into the analysis of sequences, upon which all analysis of Euclidean space (and any separable metric space) is based.
Course Information showclose
Course Designer: Clare Wickman
Primary Resources: This course comprises a range of different free, online materials. However, the course makes primary use of the following:
 University of Windsor: Elias Zakon’s Mathematical Analysis I
 University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis
 YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis
In order to “pass” this course, you will need to earn a 70% or higher on the final exam. Your score on the exam will be tabulated as soon as you complete it. If you do not pass the exam, you may take it again.
Time Commitment: This course should take you approximately 135 hours to complete. At the beginning of each unit, there is a detailed list of time advisories for each subunit. These estimates factor in the time required to watch each lecture, work through each reading thoughtfully, and complete each assignment. However, this is a very intense course, and it may take some learners much more time than others. Do not be discouraged if you exceed the time estimates! As long as you are gaining mastery of the material, you are completing the course successfully.
Tips/Suggestions: If a lecture stops making sense to you, pause it – this is a luxury you only have in an online course of this nature! – and return to the readings to get uptospeed on the material. Remember to note down the time at which you paused the lecture, in case your browser times out. Additionally, be sure to attempt to prove most of the theorems and lemmas for yourself before reading the proof given in the text. Also note how carefully each assumption has been chosen. Understanding the difference between necessary and sufficient conditions is essential for good mathematics and for doing well on the Final Exam for this course.
Learning Outcomes showclose
 use set notation and quantifiers correctly in mathematical statements and proofs;
 use proof by induction or contradiction when appropriate;
 define the rational numbers, the natural numbers, and the real numbers, and understand their relationship to one another;
 define the wellordering principle, the completeness/supremum property of the real line, and the Archimedean property;
 prove the existence of irrational numbers;
 define supremum and infimum;
 correctly and fluently perform algebraic operations on expressions involving absolute value and state the triangle inequality;
 define and identify injective, surjective, and bijective mappings;
 name the various cardinalities of sets and identify the cardinality of a given set;
 define metric spaces, open sets; define open, closed, and bounded sets; define cluster points; define density;
 define convergence of sequences and prove or disprove the convergence of given sequences;
 prove and use properties of limits;
 prove standard results about closures, intersections, and unions of open and closed sets;
 define compactness using both open covers and sequences;
 state and prove the HeineBorel Theorem;
 state the BolzanoWeierstrass Theorem;
 state and use the Cantor Finite Intersection Property;
 define Cauchy sequence and prove that specific sequences are Cauchy;
 define completeness and prove that the real line, equipped with the standard metric, is complete;
 show that convergent sequences in E are Cauchy;
 define limit superior and limit inferior;
 define convergence of series using the Cauchy criterion and use the comparison, ratio, and root tests to show convergence of series;
 define continuity; state, prove, and use properties of limits of continuous functions, including the fact that continuous functions attain extreme values on compact sets;
 define divergence of functions to infinity and use properties of infinite limits;
 state and prove the intermediate value property;
 define uniform continuity and show that given functions are or are not uniformly continuous;
 give standard examples of discontinuous functions, such as the Dirichlet function;
 define connectedness and identify connected and disconnected sets;
 construct the Cantor ternary set and state its properties;
 distinguish between pointwise and uniform convergence;
 prove that if a sequence of continuous functions converges uniformly, their limit is also continuous;
 define derivatives of real and extendedrealvalued functions;
 compute derivatives using the limit definition and prove basic properties of derivatives;
 state the Mean Value Theorem and use it in proofs;
 construct the Riemann Integral and state its properties;
 state the Fundamental Theorem of Calculus and use it in proofs;
 define pointwise and uniform convergence of series of functions;
 use the Weierstrass MTest to check for uniform convergence of series;
 construct Taylor Series and state Taylor’s Theorem; and
 identify necessary and sufficient conditions for termbyterm differentiation of power series.
Course Requirements showclose
√ have access to a computer;
√ have continuous broadband Internet access;
√ have the ability/permission to install plugins or software (e.g., Adobe Reader or Flash);
√ have the ability to download and save files and documents to a computer;
√ have the ability to open Microsoft files and documents (.doc, .ppt, .xls, etc.);
√ be competent in the English language;
√ have read the Saylor Student Handbook; and
√ have completed MA101, MA102, MA103, MA211, and MA221 from “The Core Program” in the math discipline, or their equivalents. MA111 is recommended for students lacking experience with rigorous methods of proof.
Unit Outline show close
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Unit 1: The Real Number System
In this unit, we will learn or be reminded of the tools of set theory, which underlie rigorous mathematical proof, before investigating the real numbers as a field. Beginning with the rational numbers, we will construct the real number system, a project which took mathematicians hundreds of years to fully justify. Along the way, we will encounter several fundamental concepts, including the wellordering principle, the completeness axiom, and the Archimedean property.
Unit 1 Time Advisory show close
We will see the proof of the existence of irrational numbers and learn how to use proof by induction. Finally, we will see proven a number of results about the cardinality of sets.
Unit 1 Learning Outcomes show close

1.1 Sets, Operations on Sets, and Quantifiers
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 1: Set Theory: Sections 13: Sets and Operations on Sets; Quantifiers”
Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 1: Set Theory: Sections 13: Sets and Operations on Sets; Quantifiers” (PDF)
Instructions: Please read the indicated sections, on pages 16. Please note that you will be returning to this resource throughout the course, so you may prefer to save the PDF to your desktop for quick reference.
Terms of Use: Mathematical Analysis I was written by Elias Zakon and relicensed under a Creative CommonsBy Attribution 3.0 Unported Licencse as part of the Saylor Foundation’s Open Textbook Challenge. It is attributed to the Trillia Group and Elias Zakon.  Lecture: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 1: Constructing the Rational Numbers”
Link: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 1: Constructing the Rational Numbers” (YouTube)
Instructions: Please watch this lecture, in which Professor Su discusses set and function/relation notation. He also discusses the historical development of the study of analysis and construct (that is, rigorously justify from first principles) the rational numbers.
Watching this lecture and pausing to take notes should take approximately 1 hour and 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 1: Set Theory: Sections 13: Sets and Operations on Sets; Quantifiers”

1.2 Axioms of the Real Numbers
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 2 “Real Numbers and Fields: Sections 14: Axioms and Basic Definitions”
Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 2: Real Numbers and Fields: Sections 14: Axioms and Basic Definitions” (PDF)
Instructions: Please read Sections 14, “Axioms and Basic Definitions,” on pages 2327.
Terms of Use: Mathematical Analysis I was written by Elias Zakon and relicensed under a Creative CommonsBy Attribution 3.0 Unported Licencse as part of the Saylor Foundation’s Open Textbook Challenge. It is attributed to the Trillia Group and Elias Zakon.
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 2 “Real Numbers and Fields: Sections 14: Axioms and Basic Definitions”

1.3 Integers and the Rational Numbers
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 2: Real Numbers and Fields: Section 7: Integers and Rationals”
Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 2: Real Numbers and Fields: Section 7: Integers and Rationals” (PDF)
Instructions: Please read Section 7, “Integers and Rationals,” on pages 3436. Note: The reading for this section focuses on integers and rationals, but one cannot ponder the rational numbers without considering the existence of irrationals. Hence, in the video lectures for this section, the existence of irrationals will be proven. The readings on the irrational numbers come in subunit 1.4 after a few more concepts have been developed. This difference in ordering should not interfere with your understanding of the lectures for this section.
Terms of Use: Mathematical Analysis I was written by Elias Zakon and relicensed under a Creative CommonsBy Attribution 3.0 Unported Licencse as part of the Saylor Foundation’s Open Textbook Challenge. It is attributed to the Trillia Group and Elias Zakon.  Lecture: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 2: Properties of Q” and “Lecture 3: Construction of the Real Numbers”
Link: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 2: Properties of Q” (YouTube) and “Lecture 3: Construction of the Real Numbers” (YouTube)
Instructions: Please watch both of these videos. In the first lecture, Professor Su discusses the rational numbers in further detail, defining addition and multiplication, and he will use them to introduce important concepts such as ordering. He proves the existence of irrationals, such as the square root of 2. In the second lecture, Professor Su discusses Dedekind cuts, the least upper bound property of the reals (a.k.a. the completeness property). Feel free to end the lecture at 42:00 (the discussion of Dedekind cuts).
Watching these videos and pausing to take notes should take approximately 2 hours and 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 2: Real Numbers and Fields: Section 7: Integers and Rationals”
 1.4 Upper and Lower Bounds

1.4.1 Upper and Lower Bounds
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 2: Real Numbers and Fields: Sections 89: Upper and Lower Bounds and Completeness”
Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 2: Real Numbers and Fields: Sections 89: Upper and Lower Bounds and Completeness” (PDF)
Instructions: Please read Sections 89, “Upper and Lower Bounds and Completeness,” on pages 3640.
Terms of Use: Mathematical Analysis I was written by Elias Zakon and relicensed under a Creative CommonsBy Attribution 3.0 Unported Licencse as part of the Saylor Foundation’s Open Textbook Challenge. It is attributed to the Trillia Group and Elias Zakon.
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 2: Real Numbers and Fields: Sections 89: Upper and Lower Bounds and Completeness”

1.4.2 The Archimedean Property
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 2: Real Numbers and Fields: Section 10: Some Consequences of the Completeness Axiom”
Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 2: Real Numbers and Fields: Section 10: Some Consequences of the Completeness Axiom” (PDF)
Instructions: Please read “Some Consequences of the Completeness Axiom” on pages 4346.
Terms of Use: Mathematical Analysis I was written by Elias Zakon and relicensed under a Creative CommonsBy Attribution 3.0 Unported Licencse as part of the Saylor Foundation’s Open Textbook Challenge. It is attributed to the Trillia Group and Elias Zakon.  Lecture: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 4: The Least Upper Bound Property”
Link: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 4: The Least Upper Bound Property” (YouTube)
Instructions: Please watch this video. Please note that this material covers the topics outlined in subsubunits 1.4.1 and 1.4.2. In this lecture, Professor Su further discusses Dedekind cuts and the existence of arbitrary real powers of rationals. He goes through the least upper bound property of the reals (a.k.a. the completeness property), the greatest lower bound property, and the Archimedean Property, which he proves. He proves the density of the rationals in the real line. He also gives properties of the supremum. Mastery of these topics is essential to developing a thorough understanding of analysis.
Watching this lecture and pausing to take notes should take approximately 1 hour and 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 2: Real Numbers and Fields: Section 10: Some Consequences of the Completeness Axiom”

1.5 Irrationals
 Activity: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Exercise 1.2.5: Real Numbers: The Set of Real Numbers: Exercises”
Link: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Exercise 1.2.5: Real Numbers: The Set of Real Numbers: Exercises” (PDF)
Instructions: Please work through the exercises on page 30 of this PDF. Because this resource is used by some institutions of higher learning for the purposes of assigning grades and credit for classes, complete solutions to the problems are unavailable. However, it is to your benefit to attempt the problems; your solutions should mimic the style of the proofs given in the preceding chapter.
Terms of Use: The article above is released under Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Jiri Lebl and the original version can be found here.  Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 2: Real Numbers and Fields: Sections 1112: Powers with Arbitrary Real Exponents and Irrationals”
Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 2: Real Numbers and Fields: Sections 1112: Powers with Arbitrary Real Exponents and Irrationals” (PDF)
Instructions: Please read “Powers with Arbitrary Real Exponents and Irrationals” on pages 4650.
Terms of Use: Mathematical Analysis I was written by Elias Zakon and relicensed under a Creative CommonsBy Attribution 3.0 Unported Licencse as part of the Saylor Foundation’s Open Textbook Challenge. It is attributed to the Trillia Group and Elias Zakon.
 Activity: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Exercise 1.2.5: Real Numbers: The Set of Real Numbers: Exercises”

1.6 The Natural Numbers and Induction
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 2: Real Numbers and Fields: Sections 56: Natural Numbers and Induction”
Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 2: Real Numbers and Fields: Sections 56: Natural Numbers and Induction” (PDF)
Instructions: Please read the “Natural Numbers and Induction” section on pages 2732.
Terms of Use: Mathematical Analysis I was written by Elias Zakon and relicensed under a Creative CommonsBy Attribution 3.0 Unported Licencse as part of the Saylor Foundation’s Open Textbook Challenge. It is attributed to the Trillia Group and Elias Zakon.  Lecture: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 6: Principle of Induction”
Link: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 6: Principle of Induction” (YouTube)
Instructions: Please watch this video. In this lecture, Professor Su justifies and demonstrates the principle of proof by induction. He shows that it is equivalent to the wellordering principle.
Watching this lecture and pausing to take notes should take approximately 1 hour and 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 2: Real Numbers and Fields: Sections 56: Natural Numbers and Induction”

1.7 Absolute Value
 Reading: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Section 1.3: Real Numbers: Absolute Value”
Link: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Section 1.3: Real Numbers: Absolute Value” (PDF)
Instructions: Please read Section 1.3, “Absolute Value,” on pages 3134. The most important fact we will encounter in this section, which we will use over and over again in this course and all those which follow, is the triangle inequality.
Terms of Use: The article above is released under Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Jiri Lebl and the original version can be found here.  Activity: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Section 1.3.1: Absolute Value: Exercises”
Link: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Section 1.3.1: Absolute Value: Exercises” (PDF)
Instructions: Work through the exercises on page 34. Because this resource is used by some institutions of higher learning for the purposes of assigning grades and credit for classes, complete solutions to the problems are unavailable. However, it is to your benefit to attempt the problems; your solutions should mimic the style of the proofs given in the preceding chapter.
Terms of Use: The article above is released under Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Jiri Lebl and the original version can be found here.
 Reading: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Section 1.3: Real Numbers: Absolute Value”

1.8 Intervals and the Size of R
 Reading: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Section 1.4: Intervals and the Size of R”
Link: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Section 1.4: Intervals and the Size of R” (PDF)
Instructions: Please read the indicated section, on pages 35 and 36.
Terms of Use: The article above is released under Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Jiri Lebl and the original version can be found here.
 Reading: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Section 1.4: Intervals and the Size of R”
 1.9 Sets and Countable Sets

1.9.1 Sets: Relations; Mappings
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 1: Set Theory: Sections 47: Relations; Mappings”
Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 1: Set Theory: Sections 47: Relations; Mappings” (PDF)
Instructions: Please read “Relations; Mappings” on pages 814.
Terms of Use: Mathematical Analysis I was written by Elias Zakon and relicensed under a Creative CommonsBy Attribution 3.0 Unported Licencse as part of the Saylor Foundation’s Open Textbook Challenge. It is attributed to the Trillia Group and Elias Zakon.
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 1: Set Theory: Sections 47: Relations; Mappings”

1.9.2 Countable Sets
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 1: Set Theory: Section 9: Some Theorems on Countable Sets”
Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 1: Set Theory: Section 9: Some Theorems on Countable Sets” (PDF)
Instructions: Please read “Some Theorems on Countable Sets” on pages 1821.
Terms of Use: Mathematical Analysis I was written by Elias Zakon and relicensed under a Creative CommonsBy Attribution 3.0 Unported Licencse as part of the Saylor Foundation’s Open Textbook Challenge. It is attributed to the Trillia Group and Elias Zakon.  Lecture: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 7: Countable and Uncountable Sets” and “Lecture 8: Cantor Diagonalization and Metric Spaces”
Link: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 7: Countable and Uncountable Sets” (YouTube) and “Lecture 8: Cantor Diagonalization and Metric Spaces” (YouTube)
Instructions: Please note that these lectures cover the topics outlined in subsubunits 1.9.1 and 1.9.2 of this course. Please watch each video lecture. Only watch Lecture 8 up to the 51minute mark.
In “Lecture 7: Countable and Uncountable Sets,” Professor Su revisits functions and relations between sets and important concepts such as “onetoone/injective,” “onto/surjective,” and “bijective.” He defines finite, infinite, countable, and uncountable sets and power sets, and he discusses the cardinality of certain important sets. In “Lecture 8: Cantor Diagonalization and Metric Spaces,” Professor Su discusses cardinality and Cantor’s diagonalization argument in more detail.
Watching these videos and pausing to take notes should take approximately 2 hours and 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  Activity: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Exercise 1.4.1: Real Numbers: Intervals and the Size of R: Exercises”
Link: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Exercise 1.4.1: Real Numbers: Intervals and the Size of R: Exercises” (PDF)
Work through the problems on pages 36 and 37. Because this resource is used by some institutions of higher learning for the purposes of assigning grades and credit for classes, complete solutions to the problems are unavailable. However, it is to your benefit to attempt the problems; your solutions should mimic the style of the proofs given in the preceding chapter.
Terms of Use: The article above is released under Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Jiri Lebl and the original version can be found here.
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 1: Set Theory: Section 9: Some Theorems on Countable Sets”

Unit 2: Metric Spaces
In this unit, we will learn about various topological notions and will be introduced to the more abstract notion of a metric space. A metric is a function which takes two points in a certain set and tells how “far apart” they are and which satisfies three special requirements. The metric can be used to define many different properties of the set to which it applies (the metric space). All of the concepts which you learned in calculus, especially limits, can be understood and extended in the context of metric spaces.
Unit 2 Time Advisory show close
Unit 2 Learning Outcomes show close
 2.1 Metric Spaces

2.1.1 Definition
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 3: Vector Spaces and Metric Spaces: Section 11: Metric Spaces”
Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 3: Vector Spaces and Metric Spaces: Section 11: Metric Spaces” (PDF)
Instructions: Please read the “Metric Spaces” section on pages 9598.
Terms of Use: Mathematical Analysis I was written by Elias Zakon and relicensed under a Creative CommonsBy Attribution 3.0 Unported Licencse as part of the Saylor Foundation’s Open Textbook Challenge. It is attributed to the Trillia Group and Elias Zakon.  Lecture: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 8: Cantor Diagonalization and Metric Spaces”
Link: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 8: Cantor Diagonalization and Metric Spaces” (YouTube)
Instructions: Please watch the video from the 51minute mark to the end. In this lecture, Professor Su defines metric spaces and gives examples. He defines open balls and gives examples of open balls in a variety of metrics.
Watching this lecture and pausing to take notes should take approximately 45 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  Activity: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 3: Vector Spaces and Metric Spaces: Section 11: Problems on Metric Spaces”
Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 3: Vector Spaces and Metric Spaces: Section 11: Problems on Metric Spaces” (PDF)
Instructions: Please scroll down to page 98, and work through problems 1, 9, 11, and 12.
Terms of Use: Mathematical Analysis I was written by Elias Zakon and relicensed under a Creative CommonsBy Attribution 3.0 Unported Licencse as part of the Saylor Foundation’s Open Textbook Challenge. It is attributed to the Trillia Group and Elias Zakon.
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 3: Vector Spaces and Metric Spaces: Section 11: Metric Spaces”

2.1.2 Open and Closed Sets
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 3: Vector Spaces and Metric Spaces: Section 12: Open and Closed Sets; Neighborhoods”
Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 3: Vector Spaces and Metric Spaces: Section 12: Open and Closed Sets; Neighborhoods” (PDF)
Instructions: Please read the “Open and Closed Sets; Neighborhoods” section on pages 101106.
Terms of Use: Mathematical Analysis I was written by Elias Zakon and relicensed under a Creative CommonsBy Attribution 3.0 Unported Licencse as part of the Saylor Foundation’s Open Textbook Challenge. It is attributed to the Trillia Group and Elias Zakon.  Activity: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 3: Vector Spaces and Metric Spaces: Section 12: Problems on Neighborhoods, Open and Closed Sets”
Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 3: Vector Spaces and Metric Spaces: Section 12: Problems on Neighborhoods, Open and Closed Sets” (PDF)
Instructions: Please scroll down to page 107, and work through problems 7, 8, 16, 17, and 18.
Terms of Use: Mathematical Analysis I was written by Elias Zakon and relicensed under a Creative CommonsBy Attribution 3.0 Unported Licencse as part of the Saylor Foundation’s Open Textbook Challenge. It is attributed to the Trillia Group and Elias Zakon.
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 3: Vector Spaces and Metric Spaces: Section 12: Open and Closed Sets; Neighborhoods”

2.1.3 Bounded Sets
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 3: Vector Spaces and Metric Spaces: Section 13: Bounded Sets; Diameters”
Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 3: Vector Spaces and Metric Spaces: Section 13: Bounded Sets; Diameters” (PDF)
Instructions: Please read the “Bounded Sets; Diameters” section on pages 108112.
Terms of Use: Mathematical Analysis I was written by Elias Zakon and relicensed under a Creative CommonsBy Attribution 3.0 Unported Licencse as part of the Saylor Foundation’s Open Textbook Challenge. It is attributed to the Trillia Group and Elias Zakon.  Activity: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 3: Vector Spaces and Metric Spaces: Section 13: Problems on Boundedness and Diameters”
Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 3: Vector Spaces and Metric Spaces: Section 13: Problems on Boundedness and Diameters” (PDF)
Instructions: Please scroll down to page 112, and work through problems 2, 4, 11, and 12.
Terms of Use: Mathematical Analysis I was written by Elias Zakon and relicensed under a Creative CommonsBy Attribution 3.0 Unported Licencse as part of the Saylor Foundation’s Open Textbook Challenge. It is attributed to the Trillia Group and Elias Zakon.
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 3: Vector Spaces and Metric Spaces: Section 13: Bounded Sets; Diameters”

2.1.4 Cluster Points
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 3: Vector Spaces and Metric Spaces: Section 14: Cluster Points; Convergent Sequences”
Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 3: Vector Spaces and Metric Spaces: Section 14: Cluster Points; Convergent Sequences” (PDF)
Instructions: Please read the “Cluster Points; Convergent Sequences” section on pages 114118.
Terms of Use: Mathematical Analysis I was written by Elias Zakon and relicensed under a Creative CommonsBy Attribution 3.0 Unported Licencse as part of the Saylor Foundation’s Open Textbook Challenge. It is attributed to the Trillia Group and Elias Zakon.  Lecture: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 9: Limit Points” and “Lecture 15: Convergence of Sequences”
Link: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 9: Limit Points” (YouTube) and “Lecture 15: Convergence of Sequences” (YouTube)
Instructions: Please watch these lectures. In the first lecture, Professor Su defines limit (cluster) points and goes through many examples. He also defines interior points, open sets, closed sets, and closures. In the second lecture, Professor Su defines what it means for a sequence to converge.
Watching these lectures and pausing to take notes should take approximately 2 hours and 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  Activity: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 3: Vector Spaces; Metric Spaces: Section 14: Problems on Cluster Points and Convergence”
Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 3: Vector Spaces; Metric Spaces: Section 14: Problems on Cluster Points and Convergence” (PDF)
Instructions: Please scroll down to page 118, and work through problems 5 and 10.
Terms of Use: Mathematical Analysis I was written by Elias Zakon and relicensed under a Creative CommonsBy Attribution 3.0 Unported Licencse as part of the Saylor Foundation’s Open Textbook Challenge. It is attributed to the Trillia Group and Elias Zakon.
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 3: Vector Spaces and Metric Spaces: Section 14: Cluster Points; Convergent Sequences”

2.1.5 Operations on Convergent Sequences
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 3: Vector Spaces and Metric Spaces: Section 15: Operations on Convergent Sequences”
Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 3: Vector Spaces and Metric Spaces: Section 15: Operations on Convergent Sequences” (PDF)
Instructions: Please read the indicated section on pages 120123.
Terms of Use: Mathematical Analysis I was written by Elias Zakon and relicensed under a Creative CommonsBy Attribution 3.0 Unported Licencse as part of the Saylor Foundation’s Open Textbook Challenge. It is attributed to the Trillia Group and Elias Zakon.  Lecture: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 16: Subsequences, Cauchy Sequences”
Link: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 16: Subsequences, Cauchy Sequences” (YouTube)
Instructions: Please watch this lecture through the 30minute mark. In this lecture, Professor Su further explores properties of limits of sequences.
Watching this lecture and pausing to take notes should take approximately 45 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  Activity: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 3: Vector Spaces and Metric Spaces: Section 15: Problems on Limits of Sequences”
Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 3: Vector Spaces and Metric Spaces: Section 15: Problems on Limits of Sequences” (PDF)
Instructions: Please scroll down to page 123, and work through problems 1, 10, 12, 13, and 25.
Terms of Use: Mathematical Analysis I was written by Elias Zakon and relicensed under a Creative CommonsBy Attribution 3.0 Unported Licencse as part of the Saylor Foundation’s Open Textbook Challenge. It is attributed to the Trillia Group and Elias Zakon.
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 3: Vector Spaces and Metric Spaces: Section 15: Operations on Convergent Sequences”

2.1.6 Closed Sets and Density
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 3: Vector Spaces and Metric Spaces: Section 16: More on Cluster Points and Closed Sets; Density”
Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 3: Vector Spaces and Metric Spaces: Section 16: More on Cluster Points and Closed Sets; Density” (PDF)
Instructions: Please read the “More on Cluster Points and Closed Sets; Density” section on pages 135139.
Terms of Use: Mathematical Analysis I was written by Elias Zakon and relicensed under a Creative CommonsBy Attribution 3.0 Unported Licencse as part of the Saylor Foundation’s Open Textbook Challenge. It is attributed to the Trillia Group and Elias Zakon.  Lecture: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 10: The Relationship Between Open and Closed Sets”
Link: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 10: The Relationship Between Open and Closed Sets” (YouTube)
Instructions: Please watch this lecture, in which Professor Su revisits the definition of open and closed sets and some of the subtleties involved in manipulating them. He proves standard results, such as that the closure of a set is closed and that a set is closed if and only if its complement is open. He also investigates unions and intersections of open and closed sets. He defines what it means for one set to be dense in another set.
Watching this lecture and pausing to take notes should take approximately 1 hour and 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  Activity: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 3: Vector Spaces and Metric Spaces: Section 16: Problems on Cluster Points, Closed Sets, and Density”
Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 3: Vector Spaces and Metric Spaces: Section 16: Problems on Cluster Points, Closed Sets, and Density” (PDF)
Instructions: Please scroll down to page 140, and work through problems 9, 12, and 17.
Terms of Use: Mathematical Analysis I was written by Elias Zakon and relicensed under a Creative CommonsBy Attribution 3.0 Unported Licencse as part of the Saylor Foundation’s Open Textbook Challenge. It is attributed to the Trillia Group and Elias Zakon.
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 3: Vector Spaces and Metric Spaces: Section 16: More on Cluster Points and Closed Sets; Density”

2.2 Compactness
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 6: Compact Sets and Section 7: More on Compactness”
Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 6: Compact Sets and Section 7: More on Compactness” (PDF)
Instructions: Please read the indicated sections on pages 186189 and pages 192194.
Terms of Use: Mathematical Analysis I was written by Elias Zakon and relicensed under a Creative CommonsBy Attribution 3.0 Unported Licencse as part of the Saylor Foundation’s Open Textbook Challenge. It is attributed to the Trillia Group and Elias Zakon.  Lecture: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 11: Compact Sets” and “Lecture 12: The Relationship of Compact Sets to Closed Sets”
Link: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 11: Compact Sets” (YouTube) and “Lecture 12: The Relationship of Compact Sets to Closed Sets” (YouTube)
Instructions: Click on the links above, and watch these lectures. In the first lecture, Professor Su defines compactness from the topological perspective (e.g., the way it is defined in the reading of Zakon’s Section 4.7 – every open cover of the set must have a finite subcover). This is because in his course he develops this concept before discussing the convergence of sequences. He also (in essence) defines what it means for a set to be relatively open with respect to another set. He proves that compact sets are bounded in Euclidean space. In the second lecture, Professor Su proves that compact sets are closed in Euclidean space. He proves that nested closed intervals in R have nonempty intersection. He also proves that R is uncountable.
Also, note that Professor Su will touch on sequential compactness in the lecture for subsubunit 2.3.2. In this lecture and the one that follows, he uses the opencover definition.
Watching these lecture sand pausing to take notes should take 2 hours and 45 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  Lecture: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 13: Compactness and the HeineBorel Theorem”
Link: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 13: Compactness and the HeineBorel Theorem” (YouTube)
Instructions: Please watch this lecture, in which Professor Su proves that closed, bounded intervals on the real line are compact. He then proves the HeineBorel Theorem (this is exercise 10 in section 4.6 of Zakon’s book). He states a version of the BolzanoWeierstrass Theorem (which will be discussed in the reading under subsubunit 2.3.1 for this course) and the Cantor Finite Intersection Property.
Watching this lecture and pausing to take notes should take 1 hour and 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 6: Compact Sets and Section 7: More on Compactness”
 2.3 Subsequences, Cauchy Sequences, and Completeness

2.3.1 Cauchy Sequences and Completeness
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 3: Vector Spaces and Metric Spaces: Section 17: Cauchy Sequences; Completeness”
Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 3: Vector Spaces and Metric Spaces: Section 17: Cauchy Sequences; Completeness” (PDF)
Instructions: Please read the “Cauchy Sequences; Completeness” section on pages 141144.
Terms of Use: Mathematical Analysis I was written by Elias Zakon and relicensed under a Creative CommonsBy Attribution 3.0 Unported Licencse as part of the Saylor Foundation’s Open Textbook Challenge. It is attributed to the Trillia Group and Elias Zakon.
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 3: Vector Spaces and Metric Spaces: Section 17: Cauchy Sequences; Completeness”

2.3.2 Limits Superior and Inferior and the BolzanoWeierstrass Theorem
 Reading: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Section 2.3: Limit Superior, Limit Inferior, and BolzanoWeierstrass”
Link: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Section 2.3: Limit Superior, Limit Inferior, and BolzanoWeierstrass” (PDF)
Instructions: Please read “Limit Superior, Limit Inferior, and BolzanoWeierstrass” on pages 6166.
Terms of Use: The article above is released under Creative Commons AttributionNonCommercialShareAlike 3.0 License. It is attributed to Jiri Lebl and the original version can be found here.  Lecture: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 16: Subsequences, Cauchy Sequences” and “Lecture 17: Complete Spaces”
Link: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 16: Subsequences, Cauchy Sequences” (YouTube) and “Lecture 17: Complete Spaces” (YouTube)
Instructions: Please watch both lectures. Watch “Lecture 16: Subsequences, Cauchy Sequences” from the 30minute mark to the end; watch all of “Lecture 17: Complete Spaces”. In the first lecture, Professor Su defines subsequences and proves several important results about them. Note the definition of sequential compactness at time 43:50. He proves the BolzanoWeierstrass Theorem. He defines Cauchy sequence and completeness. In the second lecture, Professor Su proves that compact metric spaces are complete. He also proves that Euclidean space is complete. He constructs the completion of a metric space. He discusses bounded sequences and monotonic sequences and proves that bounded, monotonic sequences converge. He defines limit superior and limit inferior and proves several results about them.
Watching these lectures and pausing to take notes should take approximately 1 hour and 45 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Section 2.3: Limit Superior, Limit Inferior, and BolzanoWeierstrass”

2.4 Series
 Reading: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Section 2.5: Series”
Link: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Section 2.5: Series” (PDF)
Instructions: Please read “Series” on pages 7281.
Terms of Use: The article above is released under Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Jiri Lebl and the original version can be found here.  Lecture: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 18: Series” and “Lecture 19: Series Convergence Tests, Absolute Convergence”
Link: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 18: Series” (YouTube) and “Lecture 19: Series Convergence Tests, Absolute Convergence” (YouTube)
Instructions: Please watch these lectures. In the first lecture, Professor Su defines series convergence using partial sums (a.k.a. the Cauchy Criterion). He validates the Comparison Test. He discusses the geometric series. In the second lecture, Professor Su discusses further tests for convergence, along with the definition of absolute convergence. He gives the ratio and root tests. He defines power series. He discusses summation by parts.
Watching these lectures and pausing to take notes should take approximately 2 hours and 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Section 2.5: Series”

Unit 3: Functions and Continuity
In this unit, you will explore the properties of continuous functions. You should be familiar with continuity from calculus; indeed, many of the results which are proven in this unit will be familiar, such as the intermediate value theorem or the fact that the sum of two continuous functions is continuous. However, you will now be proving these results rigorously and with more generality. You will also be learning new concepts, such as uniform continuity, and you will be improving your knowledge of what might be called “mathematical culture” through exposure to famous and muchemployed examples, such as the Dirichlet function and the Cantor Ternary set. Learn these examples well; they are used in many cases as counterexamples. Be sure also to note the use of the Triangle Inequality in the proof that the uniform limit of a sequence of continuous functions is itself continuous. This is a standard and wellknown technique.
Unit 3 Time Advisory show close
Unit 3 Learning Outcomes show close
 3.1 Functions

3.1.1 Basic Definitions
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 1: Basic Definitions”
Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 1: Basic Definitions” (PDF)
Instructions: Please read the indicated section on pages 149157.
Terms of Use: Mathematical Analysis I was written by Elias Zakon and relicensed under a Creative CommonsBy Attribution 3.0 Unported Licencse as part of the Saylor Foundation’s Open Textbook Challenge. It is attributed to the Trillia Group and Elias Zakon.
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 1: Basic Definitions”

3.1.2 General Theorems on Limits and Continuity
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 2: Some General Theorems on Limits and Continuity”
Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 2: Some General Theorems on Limits and Continuity” (PDF)
Instructions: Please read “Some General Theorems on Limits and Continuity” on pages 161166.
Terms of Use: Mathematical Analysis I was written by Elias Zakon and relicensed under a Creative CommonsBy Attribution 3.0 Unported Licencse as part of the Saylor Foundation’s Open Textbook Challenge. It is attributed to the Trillia Group and Elias Zakon.  Lecture: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 20: Functions – Limits”
Link: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 20: Functions – Limits” (YouTube)
Instructions: Please watch the entire video. In this lecture, Professor Su defines limits and limit points of functions. He defines continuity of a function using neighborhoods and using the sequential criterion. He states several properties of continuous functions, including the fact that the inverse image of an open set (under a continuous function) is open.
Watching this lecture and pausing to take notes should take approximately 1 hour and 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  Lecture: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 21: Continuous Functions”
Link: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 21: Continuous Functions” (YouTube)
Instructions: Please watch this lecture, in which Professor Su discusses the definition of continuity at a point. He proves that the inverse image of an open set under a continuous function is open and illustrates this fact with several examples. He proves that the composition of continuous functions is continuous. Finally, he proves that the forward image of a compact set under a continuous function is compact and mentions some important corollaries for realvalued functions. This lecture will cover topics you have learned in subsubunits 3.1.13.1.2.
Watching this lecture and pausing to take notes should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 2: Some General Theorems on Limits and Continuity”

3.1.3 Operations on Limits
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 3: Operations on Limits and Rational Functions”
Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 3: Operations on Limits and Rational Functions” (PDF)
Instructions: Please read the indicated section on pages 170174.
Terms of Use: Mathematical Analysis I was written by Elias Zakon and relicensed under a Creative CommonsBy Attribution 3.0 Unported Licencse as part of the Saylor Foundation’s Open Textbook Challenge. It is attributed to the Trillia Group and Elias Zakon.
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 3: Operations on Limits and Rational Functions”

3.1.4 Infinite Limits
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 4: Infinite Limits and Operations in E*”
Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 4: Infinite Limits and Operations in E*” (PDF)
Instructions: Please read the indicated section on pages 177180.
Terms of Use: Mathematical Analysis I was written by Elias Zakon and relicensed under a Creative CommonsBy Attribution 3.0 Unported Licencse as part of the Saylor Foundation’s Open Textbook Challenge. It is attributed to the Trillia Group and Elias Zakon.  Activity: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 4: Problems on Limits and Operations in E*”
Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 4: Problems on Limits and Operations in E*” (PDF)
Instructions: Please scroll down to page 181, and work through problem 4.
Terms of Use: Mathematical Analysis I was written by Elias Zakon and relicensed under a Creative CommonsBy Attribution 3.0 Unported Licencse as part of the Saylor Foundation’s Open Textbook Challenge. It is attributed to the Trillia Group and Elias Zakon.
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 4: Infinite Limits and Operations in E*”

3.1.5 Monotone Functions
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 5: Monotone Functions”
Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 5: Monotone Functions” (PDF)
Instructions: Please read the “Monotone Functions” section on pages 181185.
Terms of Use: Mathematical Analysis I was written by Elias Zakon and relicensed under a Creative CommonsBy Attribution 3.0 Unported Licencse as part of the Saylor Foundation’s Open Textbook Challenge. It is attributed to the Trillia Group and Elias Zakon.
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 5: Monotone Functions”

3.1.6 Continuity on Compact Sets
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 8: Continuity on Compact Sets and Uniform Continuity”
Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 8: Continuity on Compact Sets and Uniform Continuity” (PDF)
Instructions: Please read the indicated section on pages 194200.
Terms of Use: Mathematical Analysis I was written by Elias Zakon and relicensed under a Creative CommonsBy Attribution 3.0 Unported Licencse as part of the Saylor Foundation’s Open Textbook Challenge. It is attributed to the Trillia Group and Elias Zakon.
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 8: Continuity on Compact Sets and Uniform Continuity”

3.1.7 Intermediate Value Property
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 9: The Intermediate Value Property”
Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 9: The Intermediate Value Property” (PDF)
Instructions: Please read “The Intermediate Value Property” on pages 203209.
Terms of Use: Mathematical Analysis I was written by Elias Zakon and relicensed under a Creative CommonsBy Attribution 3.0 Unported Licencse as part of the Saylor Foundation’s Open Textbook Challenge. It is attributed to the Trillia Group and Elias Zakon.  Lecture: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 22: Uniform Continuity”
Link: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 22: Uniform Continuity” (YouTube)
Instructions: Please note that this video lecture covers the topics outlined in both subsubunits 3.1.6 and 3.1.7 of this course. Please watch this lecture, in which Professor Su recaps some basic facts about continuous functions. He defines uniform continuity and relates it to compactness, giving a number of examples. He states and proves the Lebesgue Covering Lemma (this is Theorem 1 in Section 4.7 of Zakon’s book). He proves that continuous functions map connected sets to connected sets. Finally, he proves the Intermediate Value Property.
Watching this lecture and pausing to take notes should take approximately 1 hour and 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 9: The Intermediate Value Property”

3.1.8 Discontinuous Functions
 Lecture: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 23: Discontinuous Functions”
Link: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 23: Discontinuous Functions” (YouTube)
Instructions: Please watch this lecture, which reviews material from subsubunit 3.1.5. In this lecture, Professor Su discusses some famous (or merely standard) examples of discontinuous functions, including the Dirichlet function. He discusses righthand and lefthand limits of functions. He also discusses monotone functions and why they can only have a finite number of discontinuities.
Watching this lecture and pausing to take notes should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Lecture: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 23: Discontinuous Functions”

3.2 Arcs, Curves, and Connected Sets
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 10: Arcs and Curves; Connected Sets”
Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 10: Arcs and Curves; Connected Sets” (PDF)
Instructions: Please read “Arcs and Curves; Connected Sets” on pages 211215.
Terms of Use: Mathematical Analysis I was written by Elias Zakon and relicensed under a Creative CommonsBy Attribution 3.0 Unported Licencse as part of the Saylor Foundation’s Open Textbook Challenge. It is attributed to the Trillia Group and Elias Zakon.  Lecture: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 14: Connected Sets, Cantor Sets”
Link: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 14: Connected Sets, Cantor Sets” (YouTube)
Instructions: Please watch this lecture, in which Professor Su states a few more results about compactness and constructs the Cantor ternary set. He defines perfect and connected sets. He proves that nonempty closed intervals are connected.
Watching this lecture and pausing to take notes should take approximately 1 hour and 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 10: Arcs and Curves; Connected Sets”

3.3 Sequences of Functions: Pointwise and Uniform Convergence (Z, L)
 Reading: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Chapter 6, Section 1: Pointwise and Uniform Convergence”
Link: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Chapter 6, Section 1: Pointwise and Uniform Convergence” (PDF)
Instructions: Please read the indicated section on pages 189193.
Terms of Use: The article above is released under Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Jiri Lebl and the original version can be found here.  Lecture: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 25: Taylor’s Theorem, Sequences of Functions”
Link: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 25: Taylor’s Theorem, Sequences of Functions” (YouTube)
Instructions: Please play the video from the 36minute mark to the end. In this lecture, Professor Su discusses sequences of functions and what it means for them to converge either pointwise or uniformly. He gives several classic examples of sequences of functions which converge pointwise to zero. He proves that if a sequence of continuous functions converge uniformly, their limit is continuous.
Watching this lecture and pausing to take notes should take approximately 45 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Chapter 6, Section 1: Pointwise and Uniform Convergence”

Unit 4: Differentiability and Integration
In Calculus you learned the limit definition of the derivative and connected it to a picture of a sequence of secant lines converging to a tangent line. Similarly, you learned how the integral is really the limit of a series of Riemann Sums. In this unit, you will go through the derivations of those concepts in more detail and with more regard for the technicalities. You will connect the derivative and the integral to what you have learned about sequences and series, and you will learn necessary and sufficient conditions for the exchange of limits. Much of this does in fact have important implications for numerical integration and differentiation, so it is important to understand the meaning of each of the conditions in theorems such as the Weierstrass MTest.
Unit 4 Time Advisory show close
Unit 4 Learning Outcomes show close
 4.1 Differentiation

4.1.1 Derivatives of Functions of One Real Variable
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 5: Differentiation and Antidifferentiation: Section 1: Derivatives of Functions of One Variable”
Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 5: Differentiation and Antidifferentiation: Section 1: Derivatives of Functions of One Variable” (PDF)
Instructions: Please read the indicated section on pages 251257.
Terms of Use: Mathematical Analysis I was written by Elias Zakon and relicensed under a Creative CommonsBy Attribution 3.0 Unported Licencse as part of the Saylor Foundation’s Open Textbook Challenge. It is attributed to the Trillia Group and Elias Zakon.
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 5: Differentiation and Antidifferentiation: Section 1: Derivatives of Functions of One Variable”

4.1.2 Derivatives of Extended RealValued Functions
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 5: Differentiation and Antidifferentiation: Section 2: Derivatives of ExtendedReal Valued Functions”
Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 5: Differentiation and Antidifferentiation: Section 2: Derivatives of ExtendedReal Valued Functions” (PDF)
Instructions: Please read “Derivatives of ExtendedReal Valued Functions” on pages 259265.
Terms of Use: Mathematical Analysis I was written by Elias Zakon and relicensed under a Creative CommonsBy Attribution 3.0 Unported Licencse as part of the Saylor Foundation’s Open Textbook Challenge. It is attributed to the Trillia Group and Elias Zakon.
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 5: Differentiation and Antidifferentiation: Section 2: Derivatives of ExtendedReal Valued Functions”

4.1.3 L’Hopital’s Rule
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 5: Differentiation and Antidifferentiation: Section 3: L’Hopital’s Rule”
Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 5: Differentiation and Antidifferentiation: Section 3: L’Hopital’s Rule” (PDF)
Instructions: Please read “L’Hopital’s Rule” on pages 266269.
Terms of Use: Mathematical Analysis I was written by Elias Zakon and relicensed under a Creative CommonsBy Attribution 3.0 Unported Licencse as part of the Saylor Foundation’s Open Textbook Challenge. It is attributed to the Trillia Group and Elias Zakon.  Activity: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 5: Differentiation and Antidifferentiation: Section 3: Problems on L’Hopital’s Rule”
Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 5: Differentiation and Antidifferentiation: Section 3: Problems on L’Hopital’s Rule” (PDF)
Instructions: Please scroll down to page 269, and work through problems 2, 5, and 6.
Terms of Use: Mathematical Analysis I was written by Elias Zakon and relicensed under a Creative CommonsBy Attribution 3.0 Unported Licencse as part of the Saylor Foundation’s Open Textbook Challenge. It is attributed to the Trillia Group and Elias Zakon.
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 5: Differentiation and Antidifferentiation: Section 3: L’Hopital’s Rule”

4.1.4 Mean Value Theorem
 Reading: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Chapter 4, Section 2: Mean Value Theorem”
Link: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Chapter 4, Section 2: Mean Value Theorem” (PDF)
Instructions: Please read section 4.2 on pages 135139.
Terms of Use: The article above is released under Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Jiri Lebl and the original version can be found here.  Activity: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Chapter 4, Section 2.6: Mean Value Theorem: Exercises”
Link: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Chapter 4, Section 2.6: Mean Value Theorem: Exercises” (PDF)
Instructions: Please work through problems 27 on page 140. Because this resource is used by some institutions of higher learning for the purposes of assigning grades and credit for classes, complete solutions to the problems are unavailable. However, it is to your benefit to attempt the problems; your solutions should mimic the style of the proofs given in the preceding chapter.
Terms of Use: The article above is released under Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Jiri Lebl and the original version can be found here.  Lecture: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 24: The Derivative and the Mean Value Theorem”
Link: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 24: The Derivative and the Mean Value Theorem” (YouTube)
Instructions: Please note that these video lectures address topics outlined in subsubunits 4.1.1 and 4.1.4 of this course. Watch this lecture, in which Professor Su defines the derivative and shows how to derive some of the standard rules of differentiation. He proves the existence of continuous, nowheredifferentiable functions. He states and uses the Mean Value Theorem and the Generalized Mean Value Theorem.
Watching this video and pausing to take notes should take approximately 1 hour and 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Chapter 4, Section 2: Mean Value Theorem”
 4.2 The Riemann Integral

4.2.1 The Riemann Integral
 Reading: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Chapter 5, Section 1: The Riemann Integral”
Link: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Chapter 5, Section 1: The Riemann Integral” (PDF)
Instructions: Read “The Riemann Integral” on pages 147154.
Terms of Use: The article above is released under Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Jiri Lebl and the original version can be found here.
 Reading: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Chapter 5, Section 1: The Riemann Integral”

4.2.2 Properties of the Integral
 Reading: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Chapter 5, Section 2: Properties of the Integral”
Link: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Chapter 5, Section 2: Properties of the Integral” (PDF)
Instructions: Read “Properties of the Integral” on pages 156161.
Terms of Use: The article above is released under Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Jiri Lebl and the original version can be found here.
 Reading: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Chapter 5, Section 2: Properties of the Integral”

4.2.3 The Fundamental Theorem of Calculus
 Reading: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Chapter 5, Section 3: Fundamental Theorem of Calculus”
Link: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Chapter 5, Section 3: Fundamental Theorem of Calculus” (PDF)
Instructions: Read “Fundamental Theorem of Calculus” on pages 164168.
Terms of Use: The article above is released under Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Jiri Lebl and the original version can be found here.  Lecture: YouTube: University of Nottingham: Dr. Joel Feinstein’s “Mathematical Analysis: An Introduction to Riemann Integration”
Link: YouTube: University of Nottingham: Dr. Joel Feinstein’s “Mathematical Analysis: An Introduction to Riemann Integration” (YouTube)
Also available on:
iTunes U (#13)
Instructions: Please note these videos cover the topics outlined in subsubunits 4.2.14.2.3 of this course. Please watch this lecture, in which Professor Feinstein defines characteristic functions and partitions, explains the Riemann integral, and states the fundamental theorem of calculus.
Watching this lecture and pausing to take notes should take approximately 1 hour and 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Chapter 5, Section 3: Fundamental Theorem of Calculus”
 4.3 Interchange of Limits and Series of Functions

4.3.1 Interchange of Limits
 Reading: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Chapter 6, Section 2: Interchange of Limits”
Link: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Chapter 6, Section 2: Interchange of Limits” (PDF)
Instructions: Read “Interchange of Limits” on pages 195199.
Terms of Use: The article above is released under Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Jiri Lebl and the original version can be found here.
 Reading: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Chapter 6, Section 2: Interchange of Limits”

4.3.2 Series of Functions
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 12: Sequences and Series of Functions”
Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 12: Sequences and Series of Functions” (PDF)
Instructions: Please read “Sequences and Series of Functions” on pages 227232.
Terms of Use: Mathematical Analysis I was written by Elias Zakon and relicensed under a Creative CommonsBy Attribution 3.0 Unported Licencse as part of the Saylor Foundation’s Open Textbook Challenge. It is attributed to the Trillia Group and Elias Zakon.  Lecture: University of Colorado at Colorado Springs: Professor Rinaldo Schinazi’s “Modern Analysis II, Lecture 16: Series of Functions”
Link: University of Colorado at Colorado Springs: Professor Rinaldo Schinazi’s “Modern Analysis II, Lecture 16: Series of Functions” (Flash)
Instructions: To access this lecture you will need to create a free account. Once logged in, click on the link which says “Go to the UCCS Math Video Archive.” Under “Spring Semester 2008,” click on the link to “Math 432/532,” then scroll down to Lecture 21 and click the icon on the lefthand side.
In this lecture, Professor Schinazi discusses the convergence or lack thereof of the derivatives of a sequence of functions. (In Lecture 14, not required by this course, he discussed the convergence of the integrals of a uniformly convergent sequence of functions.) He then introduces series of functions and proves the Weierstrass MTest, which is the most important result in this lecture. You need not watch the section on the Weierstrass Approximation Theorem, although it is very interesting.
Watching this lecture and pausing to take notes should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 12: Sequences and Series of Functions”

4.3.3 Absolutely Convergent Series
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 13: Absolutely Convergent Series and Power Series”
Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 13: Absolutely Convergent Series and Power Series” (PDF)
Instructions: Please read the indicated section on pages 237244.
Terms of Use: Mathematical Analysis I was written by Elias Zakon and relicensed under a Creative CommonsBy Attribution 3.0 Unported Licencse as part of the Saylor Foundation’s Open Textbook Challenge. It is attributed to the Trillia Group and Elias Zakon.  Lecture: University of Colorado at Colorado Springs: Professor Rinaldo Schinazi’s “Modern Analysis II, Lecture 21: Power Series and Ratio Rule”
Link: University of Colorado at Colorado Springs: Professor Rinaldo Schinazi’s “Modern Analysis II, Lecture 21: Power Series and Ratio Rule” (Flash)
Instructions: To access this lecture you will need to create a free account. Once logged in, click on the link which says “Go to the UCCS Math Video Archive.” Under “Spring Semester 2008,” click on the link to “Math 432/532,” then scroll down to Lecture 21 and click the icon on the lefthand side.
In this lecture, Professor Schinazi discusses power series and justifies the various convergence tests (such as the root test and the ratio test). In the second half (which is optional), he proves a result about the derivative of the pointwise limit of a sequence of differentiable functions. He later discusses termbyterm differentiation (which is really an interchange of limits).
Watching this lecture and pausing to take notes should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 4: Function Limits and Continuity: Section 13: Absolutely Convergent Series and Power Series”

4.3.4 Taylor’s Theorem
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 5: Differentiation and Antidifferentiation: Section 6: Differentials; Taylor’s Theorem and Taylor Series”
Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 5: Differentiation and Antidifferentiation: Section 6: Differentials; Taylor’s Theorem and Taylor Series” (PDF)
Instructions: Please read the indicated section on pages 288296.
Terms of Use: Mathematical Analysis I was written by Elias Zakon and relicensed under a Creative CommonsBy Attribution 3.0 Unported Licencse as part of the Saylor Foundation’s Open Textbook Challenge. It is attributed to the Trillia Group and Elias Zakon.  Lecture: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 25: Taylor’s Theorem, Sequences of Functions”
Link: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 25: Taylor’s Theorem, Sequences of Functions” (YouTube)
Instructions: Please watch this lecture up to the 36minute mark. In this lecture, Professor Su states Taylor’s Theorem and shows how it is related to the Mean Value Theorem.
Watching this lecture and pausing to take notes should take approximately 45 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 5: Differentiation and Antidifferentiation: Section 6: Differentials; Taylor’s Theorem and Taylor Series”

Final Exam
 Final Exam: The Saylor Foundation’s “MA241 Final Exam”
Link: The Saylor Foundation’s “MA241 Final Exam”
Instructions: You must be logged into your Saylor Foundation School account in order to access this exam. If you do not yet have an account, you will be able to create one, free of charge, after clicking the link.
 Final Exam: The Saylor Foundation’s “MA241 Final Exam”