Chapter 1

Wednesday

• Topic: Section 1.1: Introduction. We discuss general aims for the study of real analysis; namely, we hope to develop a rigorous understanding of the real number, convergence, and properties of single-variable real-valued functions like continuity and differentiability.
• Class materials: Lecture notes, worksheet
• After class: Read the first three pages of Section 1.2. This is material we likely won’t have time to discuss during lecture, but I’m happy to talk about in office hours or on the question and answer forum. Work on Homework 0.

Friday

• Topic: Section 1.2: Preliminaries. We discuss the triangle inequality and get refreshed on some basic proof techniques.
• Class materials: Lecture notes, worksheet
• After class: Read the first two pages of Section 1.3. Work on Homework 1.

Friday

Chapter 1

Monday

• Topic: Section 1.3: The Axiom of Completeness. We introduce the notion of the supremum of a set and the key defining property that distinguishes \(\mathbb R\) from \(\mathbb Q\).
• Class materials: Lecture notes, worksheet
• After class: Read the first three pages of Section 1.4.

Wednesday

• Topic: Section 1.4: Consequences of Completeness. We discuss some properties of the real numbers, like the Archimedean Property and the density of \(\mathbb Q\) in \(\mathbb R\), that come about due to the Axiom of Completeness.
• Class materials: Lecture notes, worksheet
• After class: Read the first two pages of Section 1.5. For tomorrow’s quiz, you should know the definitions of the upper bound of a set, the maximum of a set, and the supremum of a set. You should also be prepared to do an induction proof like the ones in Homework 0.

Friday

• Topic: Section 1.5: Cardinality. We discuss what it means for a set to be countable or uncountable.
• Class materials: Lecture notes
• After class: Read Section 2.1, which gives some good motivation for the things we’ll study in the rest of Chapter 2. Work on Homework 2.

Chapter 2

Monday

• Topic: Section 2.2: The Limit of a Sequence. We begin our discussion on limits of sequences by formally defining what it means for a sequence to converge. To help our understanding we’ll practice proving certain example sequences converge.
• Class materials: Lecture notes, worksheet
• After class: Read the first two pages of Section 2.3.

Wednesday

• Topic: Section 2.3: The Algebraic Limit Theorem. We begin our discussion on general theorems about convergent sequences. The focus today is building new convergent sequences from given ones through algebraic operations. We prove familiar facts like the limit of a sum is the sum of the limits, provided the limits exist.
• Class materials: Lecture notes, worksheet
• After class: Read the end of Section 2.3 on Limits and Order. For Friday’s quiz, make sure to know the definition for convergence of a sequence to a given value and how to find the supremum, infimum, maximum, and minimum, if they exist, of a set.

Friday

• Topic: Section 2.3: The Order Limit Theorem. We discuss inequalities involving limits of convergent sequences.
• Class materials: Lecture notes, worksheet
• After class: Read Section 2.4. Work on Homework 3.

Chapter 2

Monday

• Topic: Section 2.4: Monotone Convergence Theorem and a First Look at Infinite Series. We prove that sequences which are bounded and monotone must converge to their infimum or supremum. We also introduce the concept of infinite series and their convergence.
• Class materials: Lecture notes, worksheet
• After class: Read the first two pages of Section 2.5.

Wednesday

• Topic: Section 2.5: Subsequences. We introduce the notion of a subsequence of a sequence and show that if a sequence converges, then its subsequences must all converge to the same limit.
• Class materials: Lecture notes, worksheet
• After class: Read the proof of the Bolzano-Weierstrass Theorem in Section 2.5. For Friday’s quiz, you should know the statement of the Axiom of Completeness and the Density of \(\mathbb Q\) in \(\mathbb R\), as well as how to do proofs like in Exercises 1.3.5a and 1.4.4.

Friday

• Topic: Section 2.5: Bolzano-Weierstrass Theorem. We prove that every bounded sequence has a convergent subsequence.
• Class materials: Lecture notes
• After class: Read Section 2.6. Work on Homework 4.

Chapter 2

Monday

• Topic: Section 2.6: The Cauchy Criterion. We introduce the notion of a Cauchy sequence and prove that a sequence of real numbers converges to a real number if and only if it is a Cauchy sequence.
• Class materials: Lecture notes, worksheet
• After class: Read the first two pages of Section 2.7.

Wednesday

• Topic: Section 2.7: Properties of Infinite Series. We return to infinite series and use the theory of sequences to help us understand when infinite series converge. We prove an Algebraic Limit Theorem and Cauchy Criterion for series.
• Class materials: Lecture notes, worksheet
• After class: Read from Theorem 2.7.6 until the end of the section. For Friday’s quiz, you should know what it means for (ie. the definition of) a sequence to converge, a sequence to be bounded, and a sequence to be monotone. You should also know how to prove an example sequence converges as in Exercise 2.2.2.

Friday

• Topic: Section 2.7: Properties of Infinite Series. We discuss series convergence tests first introduced in your calculus classes.
• Class materials: Lecture notes, worksheet
• After class: Begin studying for Exam 1. Start by writing out the definitions we have learned so far and try to give an example for each. Then go over past worksheets, homework, and quizzes to get an idea of the kinds of problems to expect on the exam.

Chapter 2 and Exam 1

Monday

• Topic: Section 2.7: Properties of Infinite Series. We wrap up last week’s discussion on infinite series, focusing on Problem 2 of the worksheet from last time.
• After class: Continue studying for Exam 1.

Wednesday

• Topic: Review: We spend the day working on review problems from the worksheet below.
• Class materials: worksheet, solutions
• After class: Finish today’s worksheet and continue your review.

Friday

• Topic: Exam 1.
• After class: Work on the take-home portion of the exam.

Chapter 3

Monday

• Topic: Section 3.2: Open and closed sets. We begin our study of the topology of the real line. We introduce the notion of open sets, look at examples, and explain why arbitrary unions of open sets and finite intersections of open sets are open sets.
• Class materials: Lecture notes, worksheet
• After class: Read the subsection on Closed Sets in Section 3.2.

Wednesday

• Topic: Section 3.2: Open and closed sets. We discuss limit points, isolated points, and closed sets.
• Class materials: Lecture notes, worksheet
• After class: For Friday’s quiz, I want you to know the definitions of the following terms: open set, limit point, and closed set.

Friday

• Topic: Section 3.2: Open and closed sets. We wrap up the worksheet discussion from last time and discuss why a set is open if and only if its complement is closed.
• Class materials: Lecture notes
• After class: Enjoy your break!

Spring Break

Monday

• Topic: Spring break, no class.

Wednesday

• Topic: Spring break, no class.

Friday

• Topic: Spring break, no class.

Chapters 3 and 4

Monday

• Topic: Section 3.3: Compact sets. We discuss what it means for a set to be compact. We will see that compact sets are sets that are closed and bounded and have similar properties as finite sets.
• Class materials: Lecture notes, worksheet
• After class: Read the first three pages of Section 4.2.

Wednesday

• Topic: Section 4.2: Functional Limits. We give a precise definition for what it means for \(\lim_{x\to c} f(x)\) to exist and get practice writing proofs for example functions.
• Class materials: Lecture notes, worksheet
• After class: Read the rest of Section 4.2.

Friday

• Topic: Section 4.2: Functional Limits, continued. We continue our discussion of functional limits and relate the concept to sequences. This will give us an easy way to explain when functional limits do not exist as well as allow us to take advantage of the Algebraic Limit Theorems for sequences to prove analogous results for functional limits.
• Class materials: Lecture notes, worksheet
• After class: Work on Homework 7.

Chapter 4

Monday

• Topic: Section 4.3: Continuous Functions. We discuss the \(\epsilon\)-\(\delta\) definition of continuity of a function and the equivalent sequential criterion.
• Class materials: Lecture notes, worksheet
• After class: Finish today’s worksheet and read the first two pages of Section 4.4.

Wednesday

• Topic: Section 4.4: Continuous Functions on Compact Sets. We discuss the Extreme Value Theorem as a consequence of the idea that continuous functions map compact sets to compact sets. We also introduce the concept of uniform continuity, which is a stricter version of continuity which says the same \(\delta\) must work for all points \(c\) in the domain.
• Class materials: Lecture notes
• After class: For tomorrow’s quiz, I will give you 5 sets and for each, I will ask you to identify whether it is open and whether it is closed (in the style of Exercise 3.2.2 and 3.2.3 or Problem 2 in Homework 6). No justification will be needed, but you should be prepared to use the various ways in which we can determine these properties (the definitions; unions, intersections, and complements; or theorems like Theorem 3.2.8).

Friday

• Topic: Section 4.4: Continuous Functions on Compact Sets, continued. We give the definition of uniform continuity, study more examples of uniformly continuous functions, and give a sequential criterion for how to prove a function is not uniformly continuous.
• Class materials: Lecture notes, worksheet
• After class: Study for Exam 2. Start your study by working through past homework and worksheets, and make sure you’re familiar with all of the definitions in lecture notes.

Chapter 4 and Exam 2

Monday

• Topic: Section 4.4: Continuous Functions on Compact Sets, continued. We wrap up our discussion of uniform continuity and look at various properties of uniformly continuous functions.
• Class materials: Lecture notes
• After class: Continue studying for Exam 2.

Wednesday

• Topic: Review. We spend the day working on some problems to prepare for Exam 2.
• Class materials: worksheet, solutions
• After class: Get some rest!

Friday

• Topic: Exam 2.
• After class: Work on the take-home portion of Exam 2.

Chapter 5

Monday

• Topic: Section 5.2: Derivatives. We introduce the idea of differentiable functions and discuss some theorems that are a consequence of differentiability.
• Class materials: Lecture notes, worksheet
• After class: Finish today’s worksheet and read Theorem 5.2.6 and its proof.

Wednesday

• Topic: Section 5.3: Mean Value Theorems. We discuss the Interior Extremum Theorem, Rolle’s Theorem, and the Mean Value Theorem, a trio of fundamental results in the theory of differentiable functions.
• Class materials: Lecture notes, worksheet
• After class: Read up through Theorem 5.3.6 in Section 5.3.

Friday

• Topic: Section 5.3: Mean Value Theorems, continued. We discuss some consequences of the Mean Value Theorem, including the Generalized Mean Value Theorem and L’Hospital’s rule.
• Class materials: Lecture notes
• After class: Work on Homework 9. Read Section 6.2 up through page 176.

Chapter 6

Monday

• Topic: Section 6.2: Uniform Convergence of a Sequence of Functions. We introduce the notions of pointwise and uniform convergence for sequences of functions and work on examples.
• Class materials: Lecture notes, worksheet
• After class: Read the proof of Theorem 6.2.6.

Wednesday

• Topic: Section 6.2: Uniform Convergence of a Sequence of Functions, continued. We prove that if a sequence of continuous function converges uniformly then the limit function must be continuous. We also discuss a corresponding result for bounded functions.
• Class materials: Lecture notes
• After class: For Friday’s quiz, make sure you can prove a function is continuous or uniformly continuous on a given set using an \(\epsilon\)-\(\delta\) proof and make sure you can prove it fails to be uniformly continuous using the sequential criterion outlined by Theorem 4.4.5.

Friday

• Topic: Section 6.2: Uniform Convergence of a Sequence of Functions, continued. We discuss the connection between uniform convergence and a notion of Cauchy sequences of functions.
• Class materials: Lecture notes
• After class: Read the proof of Theorem 6.3.1.

Chapter 6

Monday

• Topic: Section 6.3: Uniform Convergence and Differentiation. We discuss the Differentiable Limit Theorem, which gives a set of conditions that guarantees that the uniform limit of differentiable functions is differentiable.
• Class materials: Lecture notes, worksheet
• After class:

Wednesday

• Topic:
• Class materials: Lecture notes, worksheet
• After class:

Friday

• Topic:
• Class materials: Lecture notes, worksheet
• After class:

Review

Monday

• Topic:
• Class materials: Lecture notes, worksheet
• After class: