Course Content

The main textbook for the class is:

• Introduction to Real Analysis by R.G. Bartle and D.R. Sherbert.

Students are strongly encouraged but not required to own a copy of this book, as we will follow its content rather closely. All problem sets will be posted below, and many exercise will be copied from the textbook (rather than, say, simply a list of exercises from the textbook).

Approximate agenda (roughly one topic per week)

1. Sets and functions.
2. Real numbers.
3. Sequences and series of numbers.
4. Limits.
5. Continuous functions.
6. Derivatives.
7. Riemann integrals.
8. Different notions of limit for sequences of functions.
9. More on series of numbers.
10. More on Riemann integrals.
11. Metric and topological spaces.

Lecture Notes

Notes by Amir Kamalian:

• January 7: Review of set theory
• January 9: Induction and finite cardinality
• January 12: Pigeonhole principle, (un)countability
• January 14: Fields
• January 16: Repeating decimals; ordering
• January 21: Ordered fields, absolute values, and some useful inequalities
• January 23: Completeness axiom
• January 28: Sequences
• January 30: Sequences continued
• February 5: Monotone convergence
• February 9: Subsequences
• February 11: Bolzano-Weierstrass Theorem
• February 13: limsup and liminf
• February 25: Cluster points and limits of functions
• February 27: Sequential limits
• March 2: Limits of functions continueds
• March 4: More results about limits
• March 6: Continuous functions
• March 9: Continuous functions: examples and nonexamples
• March 11: Continuous extension; monotonic bijections; boundedness
• March 13: Intermediate Value Theorem
• March 16: Uniform and Lipshitz continuity
• March 20: More on uniform continuity
• March 23: Limits of sequences of functions
• March 25: Uniform convergence

Homework assignments

• Problem Set 1. Due January 26.
• Problem Set 2. Due February 6.
• Problem Set 3. Due March 9.
• Problem Set 4. Due March 20 March 23.

Exams

There will be two midterm exams and one final exam. All exams will be delivered in-person. Each midterm will be 50 minutes long and worth 15% of the final grade. The final will be 2 hours long and worth 20% of the final grade.

• Midterm 1: February 2. Sample exam. Actual exam.
• Midterm 2: March 16 March 18 March 27. Sample exam.
• Final exam: April TBD.

Students may bring hand-written paper notes with them, but may not use electronic resources during the exam (except in cases approved by Student Accessibility Centre). The purpose of this is less about actually having the notes with you and more about preparing notes as a study method.