Pre-Class Assignments
Math 301 Real Analysis, Fall 2021
This page uses MathJax to display mathematical notation, so please let me know if any part isn't clear.
All section numbers refer to Understanding Analysis, Second Edition by Abbott.
Be sure to check back, because this page will be updated often during the semester.
Week 1: August 30
Welcome to Real Analysis!
To Read
- Mathematics for Human Flourishing by Francis Su, posted to onCourse
- Section 1.1 The Irrationality of \( \sqrt{2} \)
- Section 1.2 Some Preliminaries (skim)
- Section 1.3 The Axiom of Completeness
- Section 1.4 Consequences of Completeness (through Theorem 1.4.3)
The first reading is a short article that I think every math major should read and think about.
Section 1.2 should mostly be review from Discrete Math.
Pre-Class Forum Posts
- Due Wednesday 9/1 @ midnight
What is your reaction to Mathematics for Human Flourishing? Are there any parts that you really connected with? Post your reaction to the onCourse forum "Class discussions about readings" - Due Wednesday 9/1 @ midnight
Fill out the Bio Sheet Google form linked from onCourse
Week 2: September 6
What distinguishes \( \mathbb{Q} \) and \( \mathbb{R} \)?
How do you compare the size of infinite sets?
To Read
- Section 1.4 Consequences of Completeness (finish this section)
- Section 1.5 Cardinality (through Example 1.5.4)
Pre-Class Forum Posts
- Due Monday 9/6 @ midnight
Submit at least one comment/question to the onCourse forum "Class discussions about readings"
- Due Wednesday 9/8 @ midnight
Respond to at least one comment/question that another student posted in the forum
Week 3: September 13
\( \mathbb{R} \) and \( \mathbb{Q} \) have different cardinatlities
To Read
- Section 1.5 Cardinality (finish this section)
Pre-Class Forum Posts
- Due Monday 9/13 @ midnight
Submit at least one comment/question to the onCourse forum "Class discussions about readings"
- Due Wednesday 9/15 @ midnight
Respond to at least one comment/question that another student posted in the forum
Week 4: September 20
How many different infinities are there?
\( P(\mathbb{N})\sim \mathbb{R} \)
To Read
- Section 1.6 Cantor's Theorem
Don't try to complete all the exercises sprinkled through this section, but focus on understanding the statements of Theorem 1.6.1 and 1.6.2.
Pre-Class Forum Posts
- Due Monday 9/20 @ midnight
Submit at least one comment/question to the onCourse forum "Class discussions about readings"
- Due Wednesday 9/22 @ midnight
Respond to at least one comment/question that another student posted in the forum
Week 5: September 27
Algebraic and topological definitions of convergence
To Read
- Section 2.2 The Limit of a Sequence
- Section 2.3 The Algebraic and Order Limit Theorems
- Section 2.4 The Monotone Convergence Theorem
Read these sections, focusing on the definitions and statements of theorems, but no forum posts this week due to Exam 1.
Week 6: October 4
Every bounded sequence has a convergent subsequence
An equivalent condition for convergence
To Read
- Section 2.5 Subsequences and the Bolzano-Weierstrass Theorem
- Section 2.6 The Cauchy Criterion
The subsection "Completeness Revisited" in Section 2.6 is a nice summary of how many of the results to this point fit together.
Pre-Class Forum Posts
- Due Monday 10/4 @ midnight
Submit at least one comment/question to the onCourse forum "Class discussions about readings"
- Due Wednesday 10/6 @ midnight
Respond to at least one comment/question that another student posted in the forum
Week 7: October 11
No class meetings this week due to Fall Break and MAP Day.
Week 8: October 18
Thow away the middle third
Introduction to the topology of ℝ
To Read
- Section 3.1 The Cantor Set
- Section 3.2 Open and Closed Sets
Pay special attention to the definition of a "limit point of A" and an "isolated point of A" in Section 3.2.
Pre-Class Forum Posts
- Due Monday 10/18 @ midnight
Submit at least one comment/question to the onCourse forum "Class discussions about readings"
- Due Wednesday 10/20 @ midnight
Respond to at least one comment/question that another student posted in the forum
Week 9: October 25
Limit points on closed sets
That's a continuous function?
To Read
- Section 3.3 Compact Sets
- Section 4.1 Examples of Dirichlet & Thomae
- Section 4.2 Functional Limits
Pre-Class Forum Posts
- Due Monday 10/25 @ midnight
Submit at least one comment/question to the onCourse forum "Class discussions about readings"
- Due Wednesday 10/27 @ midnight
Respond to at least one comment/question that another student posted in the forum
Week 10: November 1
Consequences of continuity
To Read
- Section 4.3 Continuous Functions
- Section 4.4 Continuous Functions on Compact Sets
Pay special attention to the definition of a "uniform continuity" in Section 4.4.
Pre-Class Forum Posts
- Due Monday 11/1 @ midnight
Submit at least one comment/question to the onCourse forum "Class discussions about readings"
- Due Wednesday 11/3 @ midnight
Respond to at least one comment/question that another student posted in the forum
Week 11: November 8
The IVT is "obvious" but slippery to prove
To Read
- Section 4.5 The Intermediate Value Theorem
- Section 5.1 Are Derivatives Continuous?
- Section 5.2 Derivatives and the Intermediate Value Property
Pre-Class Forum Posts
- Due Monday 11/8 @ midnight
Submit at least one comment/question to the onCourse forum "Class discussions about readings"
- Due Wednesday 11/10 @ midnight
Respond to at least one comment/question that another student posted in the forum
Week 12: November 15
The MVT is "obvious" but slippery to prove
A continuous function with "corners" everywhere
To Read
- Section 5.3 The Mean Value Theorem
- Section 5.4 A Continous Nowhere-Differentiable Function
Read these sections, and focus on the definition of g(x) in Section 5.4, but no forum posts this week due to Exam 2.
Week 13: November 22
Uniform convergence is your friend
To Read
- Section 6.1 Discussion: The Power of Power Series
- Section 6.2 Uniform Convergence of a Sequence of Functions
Pre-Class Forum Posts
- Due Monday 11/22 @ midnight
Submit at least one comment/question to the onCourse forum "Class discussions about readings"
No follow-up response required due to Thanksgiving Break.
Week 14: November 29
Uniform convergence is really your friend
To Read
- Section 6.3 Uniform Convergence and Differentiation
- Section 6.4 Series of Functions
- Section 6.5 Power Series
Pre-Class Forum Posts
- Due Monday 11/29 @ midnight
Submit at least one comment/question to the onCourse forum "Class discussions about readings"
- Due Wednesday 12/1 @ midnight
Respond to at least one comment/question that another student posted in the forum
Week 15: December 6
Polynomials are a universal gadget for continuous functions
To Read
- Section 6.7 The Weierstrass Approximation Theorem
Pre-Class Forum Posts
- Due Monday 12/6 @ midnight
Submit at least one comment/question to the onCourse forum "Class discussions about readings"
- Due Wednesday 12/8 @ midnight
Respond to at least one comment/question that another student posted in the forum