Math 302: Real Analysis

Course Content

The course is divided into the following seven units. 

Unit 1: Constructing the Real Numbers

This unit begins with a lesson on proofs, and then proceeds to begin the journey towards constructing the set of real numbers. Along the way, we study several proof types (e.g., proof by contradiction) and practice using them to prove facts about natural numbers, rational and irrational numbers, and the set of real numbers. 

Unit 2: Functions and Countability

In this unit we formalize the notion of a function familiar from precalculus, and then introduce a concept -- countability -- that helps us compare the sizes of two sets ("size" understood as "number of elements in"), called the cardinality of a set. One surprising consequence of this development: while the natural numbers, integers, and rational numbers all have the same cardinality, the set of real numbers has a different cardinality. 

Unit 3: Euclidean Space and Topology

This unit studies the geometry and topology of R^n, n-dimensional Euclidean space. We begin by studying R^n as a metric space (a space equipped with a measure of distance between two points), and then move on to using the "natural" metric (the Pythagorean distance familiar from R^2) to define open n-balls. We then study R^n as a topological space, with those n-balls forming the "basis" of that topology. Later lessons then use these concepts to generalize the notion of open and closed intervals to the setting of R^n, and study various limit-like notions that appear as topological concepts (e.g., accumulation points). The last four lessons focus on compactness, which generalizes to R^n the properties of closed intervals in R that are foundational to some of the main results in calculus (e.g., the Extreme Value Theorem). 

Unit 4: Sequences

This unit begins to apply the topological concepts developed in Unit 3 to infinite sequences, a familiar concept in calculus and, as we'll see in the next unit, a foundational concept in real analysis. After defining sequences, we'll connect the limit point of a sequence (the point a sequence converges to) to adherent points and accumulation points. We'll then end by discussing Cauchy sequences and their relationship to completeness and compactness.

Unit 5: Limits and Continuity

This unit starts the journey of rigorizing calculus in earnest. First, we define the notions of limit and continuity, and do so from various perspectives (e.g., the sequential perspective; the topological perspective). We then proceed to make rigorous (and generalize) the Extreme Value Theorem. We also relate the results to a new topological concept called homeomorphisms. Next, we rigorize the Intermediate Value Theorem. In so doing, we introduce the notion of connectedness to help us describe the ``doesn't skip any of its image values'' conclusion of the Intermediate Value Theorem. We end with a lesson on uniform continuity, a special type of continuity that will help us rigorize Riemannian integration in Unit 7.

Unit 6: Differentiation

This penultimate unit we'll study differentiation. Starting with the limit definition of the derivative, we'll prove various facts about derivatives you learned in calculus. We'll then connect the derivative to local and global extrema, and prove an important theorem we'll later use to prove the Fundamental Theorem of Calculus: the Mean Value Theorem.

Unit 7: Riemann Integration

This final unit in the course rigorizes the integration portion of Calculus 1. We'll begin by defining what it means to be integrable and then proving the Integrability Criterion, which converts the definition of integrability into a condition involving epsilon. Then, in Lesson 27 we'll use that Criterion -- along with prior results we've proven in the course -- to prove the Fundamental Theorem of Calculus, thereby connecting integration and differentiation.