MAT 127A - Real Analysis
Course website for an introductory course in real analysis in Fall 2025
Homework Solutions (Joint with Chen Liang)
Homework 1 Solutions due Wednesday, October 8th, 2025
Homework 2 Solutions due Monday, October 13th, 2025
Homework 3 Solutions due Wednesday, October 29th, 2025
Homework 4 Solutions due Monday, November 10th, 2025
Homework 5 Solutions due Tuesday, November 25th, 2025
Homework 6 Solutions due Sunday, December 7th, 2025
Lecture Notes
[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!
[10/16, 4:00pm-6:00pm] Discussion 4 Notes
We first warm-up with a problem where we can practice using the definition of supremum as a least upper bound. In it we find an alternative characterization of the supremum of a set as an element with the property that if you add to it an arbitrary small constant, you get an upper bound for the set, and if you subtract from it an arbitrary small constant you do not have an upper bound. This equivalence is nice for geometric insights! We then move to recognizing whether certain subsets of the reals is dense. In particular, we see that often, we need fairly special circumstances to rigorously prove a subset of the reals is dense. The geometric idea is that around any point in the subset you can find other elements in the subset which are arbitrarily close to it. We end with a fun exposition about the construction of the real numbers due to Dedekind! I think this proof is great as we see pretty much ALL of the previous developments in set theory and suprema we had built up for the past month or so in one proof!
[10/23, 4:00pm-6:00pm] Discussion 5 Notes
The warm-up discusses the uniqueness of limits of convergent sequences. Though it is intuitively obvious, apriori there is no reason to conclude that if a sequence converges to one value, it cannot also converge to another, however, this is just a nice and quick gut check. The main substance of these notes is to provide a clean and ubiquitous way to approach convergence proofs. In particular, I prefer to do scratchwork which makes the quantity $|a_{n}-L|$ simpler and reason for what a sensible value of $N$ ought to be. We do many examples with a wide variety of difficulty, however, I have tried to collect all the examples of different methods which are practical for this course level. Though there is a preview of more formal properties about sequences and subsequences, we will wait to talk deeply about them until next week!
[10/30, 4:00pm-6:00pm] Discussion 6 Notes
We discuss the subtleties of the main theorems about sequences and subsequences. Starting with a subtlety about the order limit theorem: it does not necessarily preserve strict inequalities! We then discuss the implications of monotone convergence theorem; in particular, in the setting of monotone sequences, convergence and boundedness are equivalent conditions! This is not the case for arbitrary sequences; namely convergence implies bounded, but not necessarily the other way around. After discussing some enlightening "proof or counterexample" problems, we move to a discussion of the Bolzano-Weierstrass theorem. It is extremely important to note that this theorem is not an if and only if. Moreover, it is just an existence result, there is no way to conclude things about general subsequences of a given bounded sequence using this theorem. We give a very interesting proof of the theorem which differs from the one in class and uses ideas of supremum to review previous course content. Finally we do a quick proof involving algebraic properties about Cauchy sequences to prime for the homework.
[11/6, 4:00pm-6:00pm] Discussion 7 Notes
In this discussion we start with how to geometrically approach proving subsets of the real numbers is open, closed, both, or neither. For this, the neighborhood definitions of the above notions are very helpful. If you do not have an explicit description of the set, but you do have a topological description of various related sets, using the complement characterizations of open and closed as well as their interactions with unions and intersections, we can often find very quick and efficient proofs of openness and closedness.
[11/13, 4:00pm-6:00pm] Discussion 8 Notes
In this discussion, we discuss some grading decisions on the midterm, then move to an example problem where the main advantages of the open cover compactness characterization can shine. We show that locally constant functions on $[0,1]$ are globally constant on $[0,1]$. This problem captures the two main ideas of open cover compactness: 1) The local-to-global process and 2) The transfer of properties of finite sets to topological properties.
[11/20, 4:00pm-6:00pm] Discussion 9 Notes
We start with a discussion of functional limit proofs and relate the process to that which we did for convergence proofs in the context of sequences. We see that much the the philosophy to the approach is the same, but the explicit computations require some more cumbersome care. It is often the case in convergence proofs that you may need to take the maximum of a finite set of "cut-off points" $N_{1},N_{2},..., N_{k}$ to conclude easier. Functional limits will take advantage of minimums in these scenarios as we will see. The fundamental reason is that in the context of convergence we are looking for natural numbers $N$ for which some condition holds for all larger natural numbers $n$. In ths context of functional limits, we are looking for the opposite kind of "cut-off"; i.e. a $\delta$ for which some condition holds for all $x$ in the domain with $|x-c| < \delta$.
[12/4, 4:00pm-6:00pm] Discussion 10 Notes
We end the quarter with discussions about uniform continuity. We compare the epsilon delta definitions of continuity and uniform continuity to make sure there is a clear distinction of order of logical quantifiers and compute some examples. Then, we compare the negations of continuity and uniform continuity with sequences and point out the key detail that sequences chosen in the uniform continuous setting need not be convergent! Finally, we discuss the "moral" of uniform continuity capture in the slogan: "uniform continuity = slow/no blow up on the domain of choice". After a few examples and discussing features of two sufficient criterion for uniform continuity we see where this slogan comes from.