How I got my first A+ at Columbia

Even though mathematics is part of my major, it has never been the subject that came most easily to me. I have often found math intimidating, especially when I am starting something new and do not yet know how the pieces fit together. Calculus and linear algebra, which are more computational, were both a grind for me.

Real analysis was difficult in a different way. Introduction to Modern Analysis is Columbia’s version of a standard real analysis course, following the first seven chapters of what’s colloquially known as “Baby Rudin” — Rudin’s Principles of Mathematical Analysis, 3rd edition. The course is built around definitions, theorems, and proofs. It is less about carrying out a calculation than understanding a statement precisely enough to explain why it is true.

I knew the course’s reputation before I enrolled. Several friends who are much stronger in mathematics and economics than I am had struggled with Modern Analysis at Columbia. The class itself reflected that reputation: nearly everyone was either preparing to apply to graduate school or was already enrolled in a graduate program.

I ended up earning my first A+ at Columbia. There was no single trick behind it, and some of what I did was probably more than necessary. But I also did not enter the course with effortless confidence in math. I had to prepare deliberately.

Why I cared about real analysis

For economics students considering a PhD, real analysis is one of the courses admissions committees can recognize fairly consistently across institutions. A strong grade does not settle an admissions decision, but it gives a committee evidence that a student can work with definitions, abstraction, and proof-based arguments. The American Economic Association’s guidance on mathematical preparation includes real analysis explicitly. For a less formal sense of how much applicants worry about the course, there is no shortage of discussion online.

The course also matters for reasons beyond signaling. Many ideas that appear throughout economics depend on analysis. If a continuous objective function is defined on a compact choice set, the extreme value theorem gives conditions under which an optimum exists. Consistency in econometrics is a statement about convergence. Fixed-point results are often used to establish the existence of an equilibrium. Efe Ok’s Real Analysis with Economic Applications is devoted to these connections.

Knowing that made the course feel less like an admissions ritual. I could see why the material would matter later, even when I was struggling with it in the moment.

I attended the lectures before I enrolled

I began a year in advance, although “studying” may be too generous a word for what I was doing at first. I sat in on the lectures, recorded them, and tried to follow along.

Most of the time I felt like a deer in the headlights. The material was abstract, the notation was unfamiliar, and it often seemed as though everyone in the room was speaking a language I had never learned. I understood very little.

I was not expecting to. The point was to get some early exposure to the vocabulary and structure of the course. When I encountered the same ideas again after formally enrolling, they were still difficult, but they were no longer completely new.

I also watched outside lectures. The resource I used most was Francis Su’s Spring 2010 real analysis course at Harvey Mudd College. It was filmed with a potato, but it was worth every pixel. Su is unusually good at explaining why a definition is formulated the way it is, rather than presenting it as something the student simply has to accept.

I tried not to let confusion accumulate

My roommate at the time, a math genius who is about to begin a PhD in Business Economics at MIT Sloan, gave me one rule: do not move on until you understand what you are reading. Not only the general idea, but the actual statement in front of you.

I took that seriously. I would stop at a paragraph or even a sentence until I could explain what it meant in my own words. Recognizing the notation was not enough. I wanted to know what each assumption was doing and why the conclusion followed from it.

This made studying very slow. Given my shaky foundation, I had to go backward constantly. Sometimes I would reach a later theorem and realize that I had been relying on an earlier definition without really understanding it. Then I would return to the earlier section and work through it again.

That recursive process was frustrating, but it prevented small gaps from quietly becoming larger ones. Real analysis is cumulative. A vague understanding of limits eventually becomes a vague understanding of continuity, compactness, or convergence, and by then it can be hard to identify where the confusion began.

I read before class

Before each lecture, I read the relevant chapter and tried to familiarize myself with the main definitions and results. I was not trying to master the material before class. I wanted enough familiarity that the lecture could reinforce something I had already seen.

Walking into class cold placed too much pressure on the lecture itself. A professor has to teach a room full of students who arrive with different backgrounds and different points of confusion. Even a very good professor cannot calibrate every explanation to the exact place where I am stuck.

This mattered even more because the course met from 5:30 to 8:40 p.m. By the end of a three-hour evening lecture, I was not always in the best condition to absorb a completely new abstract idea. Reading beforehand meant I could use the lecture to clarify and organize the material rather than encounter all of it for the first time.

I taught the material to other people

When my study group met, I tried not to be learning the topic for the first time. I prepared enough beforehand that I could explain my understanding, answer questions, and notice where my explanation broke down.

Real analysis is especially useful to discuss aloud because the formal notation can hide the underlying logic. The epsilon-delta definition of a limit is a good example. Written compactly, it can look like a wall of quantifiers. Spoken plainly, it becomes more manageable: someone chooses a tolerance, however small, and your job is to find a distance from the point that keeps the function within that tolerance.

Explaining a definition this way does not replace the formal statement. It gives the formal statement a structure you can hold onto. Teaching also made it difficult for me to pretend I understood something when I did not. If I could not explain why a step was valid, I had found the part I needed to study again.

I rewrote the course in my own notes

Professor Carneiro provided notes, and previous teaching assistants had also tried to compile and organize the material. I still found that I was missing context in places, especially when a proof on the board relied on an earlier idea that I had not yet connected to it.

After each lecture, I translated the board work into my own set of notes. I kept the organization consistent and hierarchical, and I added comments, remarks, and visual aids wherever I thought the original presentation moved too quickly for someone with my background.

I also recorded the lectures and studied the transcripts afterward. Admittedly, when I am tired, my memory can be like a goldfish’s. The recordings let me recover explanations that made sense in the room but disappeared from my memory by the next morning.

Writing the notes became part of how I studied. To organize a theorem properly, I had to decide which definitions came first, which earlier results it used, and where a reader was likely to need more explanation. The notes are available at real-anal.tylersotomayor.com, and I am still adding to them.

I looked for recurring proof strategies

I am still not especially good at writing proofs or mathematical arguments. That only improves with practice. What helped me was to stop treating every proof as if it had been invented from nothing and start looking for recurring modes of argument.

A uniqueness proof, for example, often begins by assuming that two objects satisfy the required conditions and then showing that they must be equal. Many limit arguments use the triangle inequality and divide a desired error bound into smaller pieces. Contradiction, contraposition, induction, and direct construction each have situations in which they appear repeatedly.

Recognizing these patterns did not tell me the whole proof. It gave me somewhere to begin. Instead of facing a blank page, I could ask what kind of statement I was trying to prove, what the hypotheses gave me, and which familiar argument might connect them to the conclusion.

What worked for me

A lot of this may be overkill. I do not think everyone needs to attend a course a year before enrolling in it or transcribe every lecture. These were the methods that worked for me because I needed repeated exposure and more time with the definitions than some of my classmates did.

The parts I would recommend most strongly are simpler: read before class, do not leave basic definitions half-understood, explain the material to someone else, and treat proof writing as a skill that improves through repeated practice rather than as evidence of fixed mathematical talent.

Many of my friends have taken analysis, are taking it now, or plan to take it soon. That is why I plan to keep building the notes into a larger resource, with additional lectures, references, and possibly some interactive aids. The current version is available here.