MAT 127A - Real Analysis
Course website for an introductory course in real analysis in Fall 2025
Homework
Homework 1 Solutions due Tuesday, October 8th, 2025
Homework 2 Solutions due Tuesday, October 8th, 2025
Lecture Notes
WEEK 1: Review of Proof Techniques and Examples
[9/25, 4:00pm-6:00pm] Discussion 1 Notes
Important logistics discussed. We reviewed the four main proof techniques we will be using throughout the course: Direct proof, proof by contradiction, proof by contraposition, and proof by induction. We go through a handful of instructive examples and discuss how to choose between the techniques.
[10/2, 4:00pm-6:00pm] Discussion 2 Notes
We warm up by pointing out a subtle, yet common misconception about applying induction arguments; in particular, it allows you to prove a countable number of statements simultaneously, but it does not let you prove a statement involving countably many operations. We use the infinite version of De Morgan's laws to make this point. We then discuss functions of sets. First we gain a sense for injectivity and surjectivity via examples of familiar functions from calculus, then we discuss the important interactions of composition and 'jectivity. In particular, though compositions of 'jective functions remain 'jective, it is not the case that if a composition of two functions is 'jective, that each function inividually is 'jective. This is a common mistake of undergrad mathematicians! Though there are some partial reverse statements that can be proven. There are optional sections on left and right inverses and 'jectivity (particularly useful for beginner category theorists/algebraists) and 'jectivity in linear algebra for those who are interested. Finally, we bare the fruits of our labor and we use the above intuition to prove cardinality results! There are MANY different and useful techniques to prove cardinality results of sets, I included most of the reasonable ones in increasing level of difficulty to apply and I dub these examples "Zach's Ladder of Cardinality Proofs" as once these examples are known to heart, most examples will be!
[10/9, 4:00pm-6:00pm] Discussion 3 Notes
We warm up by generalizing the proof that $\sqrt{2}$ is irrational to cover all bases. We make use of Euclid's unique prime decomposition of integers to conclude that $\sqrt{n}$ is rational if and only if $n$ is a perfect square! We then pick up from the previous week's discussion about cardinality which got cut somewhat short. I introduce the fact that if $X$ is an uncountable set, and $A$ is a countable subset then $X \setminus A$ is uncountable! Thus, in a sense uncountable sets are "too far gone" and can't be made countable by removing countable subsets. We switch gears to prove elementary results about suprema and use the Archimedian Principle to help prove that a number is the supremum of various explicit sets; many involve techniques which are quite involved. We finish with some examples of formal proofs with suprema of subsets of the real numbers.