Important!

To follow the course, you must ****be familiar with basic set theory, supremum and infimum, limsup and liminf, and limits. Many freely available analysis textbooks cover these topics, for example Lectures 1—9 of the MIT lecture notes by Dote and Rodriguez and I strongly recommend reviewing this material if you don’t remember parts of it. Below is a more detailed list of prerequisites, but the four topics listed above are essential.

Set theory

• Basic logic and quantifiers
• Induction
• Sets, subsets, power set of a set
• Operations on sets: union, intersection, difference, complement, de Morgan’s laws
• Maps between sets: domain, codomain, image, preimage, injective\surjective\bijective maps
• Countable and uncountable sets, cardinality
• Cartesian product

Analysis

• Supremum and infimum
• Sequences, limits, series, Cauchy sequences
• Limsup and liminf
• Limits and continuity of functions, uniform continuity
• Pointwise and uniform convergence of functions
• Metric spaces, open and closed sets, interior and closure
• (Optional but helpful: Topological spaces)