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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 real-valued 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

Welcome to MA241: Real Analysis I. General information about this course and its requirements can be found below. In many ways, this course is the true gateway into the mathematics major, requiring rigorous proofs, introducing important topological concepts, and laying the groundwork for most of the remaining courses you will take as a math major. The precise mathematical definitions introduced in this course are used by many fields at the graduate level, including Economics, Physics, and Electrical Engineering. As you progress through the material, stop to reflect on each theorem or definition. Look for its motivation behind the introduction of each new concept – remember, these definitions were written by people who wished to use them for a purpose! – and try to work through each proof. Mathematics is not a spectator sport! Taking care at this time to understand the material deeply will pay dividends throughout your study of mathematics.

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:
Requirements for Completion: In order to complete this course, you will need to work through each unit and all of its assigned materials. 
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 up-to-speed 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

Upon successful completion of this course, you will be able to:
  • 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 well-ordering 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 Heine-Borel Theorem;
  • state the Bolzano-Weierstrass 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 extended-real-valued 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 M-Test to check for uniform convergence of series;
  • construct Taylor Series and state Taylor’s Theorem; and
  • identify necessary and sufficient conditions for term-by-term differentiation of power series.

Course Requirements  showclose

In order to take this course, you must:

√    have access to a computer;

√    have continuous broadband Internet access;

√    have the ability/permission to install plug-ins 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 MA101MA102MA103MA211, 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.

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