Real Analysis: A survival guide
It is coming for you
If you’re reading this you likely will take/are taking real analysis. In a sentence, this course is about formalizing calculus. You will see a lot of the same topics from calculus I & II, but now you’re doing proofs.
I struggled mightily in this class over the semester — it was easily my hardest class in college so far, and it came down to a do-or-die final exam to get the grade I wanted. So having clawed my way through the class, I think I made enough mistakes that I definitely know what not to do, as well as how to succeed.
Why should I try?
Aside from your GPA, and the fact that it looks nice on a transcript, what’s the point of learning hard math with no apparent applications? Will you be asked to determine if a sequence is Cauchy or not in your future job? Probably not. But will the general mathematical principles you learn in analysis be useful in your line of work?
Again, probably not. The real value add of real analysis is more subtle, but very important and transferable. In my opinion, it is not so much about building mathematical skills as it is formalizing your thinking, and making rigorous argument. Analysis will help you understand exactly what conditions an assertion depends on, how to build intricate arguments based off of a set of assumptions, and how to think more precisely. I think this course is one that actually makes your smarter, or at least a more disciplined thinker.
Anyways, without further ado, here’s 5 tips to survive real analysis.
Rule #1: Don’t psyche yourself out
I think it is fairly common for people to regard analysis as a subject where success is completely based off of mathematical intuition, and you either “have it or you don’t.” Although certain people definitely seem to have a knack for it (the kid who got perfect scores on all the tests,) this isn’t really true. A good “math sense” is developed through practice — many smart proofs and solutions arise because you might have seen a similar problem before. Don’t let “not being a genius” discourage you.
Rule #2: Start the homework early
I know that every class has the same generic advice “don’t start the homework late, start early” but let’s be real, you’re in college, you live on the Sunday night Monster induced grind session that ends at 11:58. You can hash out diff-eq homework, or cough up an essay, or blitz out a program the night before it is due.
But real analysis is different. I started a homework late, and I got a 10 on it. Not because I didn’t finish, I just actually got 10/100 points on the homework. It was demoralizing.
The reason is that the homework in this course is often just 6–10 problems that all require proofs, and to solve each problem you need to “get” it. If you have all the right insights about all the problems, you could probably crank out the homework in an hour, but good luck with that. Another problem is that you can’t just scribble out your work like you would in calculus — I had to rewrite a lot of my proofs to make them nice and readable, which takes time.
Rule #3: Do more than just attending lecture
I honestly don’t know how people are able to learn math from just lectures. Like I don’t know how someone goes to a math class, listens to the lecture, and then just internalizes the information. This may make math Professors sad, but I almost never learn the concept from the lecture alone — it takes a while for that information to sit in my brain before I can really process it.
That’s why I am going to suggest something that is hard, but I am 100% sure is the easiest way to do well in real analysis — read ahead. Just read one section ahead, and just enough to familiarize yourself with the basics of the next concept. Know the delta epsilon definition for continuous functions before you start the chapter on continuity, know what uniform convergence is before you dive into it. That way, you can really follow along in your Professor’s lecture, and you won’t just tap out in the beginning where you can’t follow with the first key concept they cover.
Don’t do what I did, where you just be confused during lecture and relearn everything later with the textbook — it’s a lot of work.
Rule #4: Don’t lie to yourself
One thing that makes analysis so hard is that the topics you cover seem simple. You’re covering everything you have seen before; limits, convergence, functions, and so on, just with more depth. And it is really easy to convince yourself that you understand a certain concept when you actually don’t — like what actually is a limit? When you take n > N for convergence, what do each of those n’s stand for? Why is the Archimedean principle a necessary condition to prove the density of the rational numbers? I struggled early in the course because I hadn’t fully grasped these simple concepts, which caused me a lot of pain down the road. Real analysis is a course that conceptually builds on itself, so your basics have to be solid. You have to be willing to ask questions that you think are stupid (what is convergence?) if you don’t 100% understand the details.
Real analysis is a course that is deceptively simple — there’s not a lot of difficult computation, but the class will pry and tug at your understanding of simple concepts to make sure it is airtight.
Rule#5: Talk to others
I think part of the reason this class was hard for me is that it was just me vs the world. Talk to your Professor — go to their office hours, and start going at the beginning of the semester. This will make sure that someone is tracking your understanding of key ideas, and ensure your conceptual foundation doesn’t have gaps or holes. If I had to do it again, I would just start the homework at the beginning of the week, and then go to my professor to ask for help with a homework question I was struggling on. That way they could help you with the homework, and clear up any confusion you had.
A buddy would also be nice in the class, but honestly, I think you need someone to help you who just knows what they are doing, so they know where and why you went wrong. Working with a classmate would also be nice because you could share the pain of getting destroyed by a test as well.
Good Luck
You’re going to need it. I found real analysis to be a difficult, but fascinating course. If you enjoy this class, it probably means you are going to enjoy mathematics in general. If you don’t enjoy this class, then at least you know you can test yourself with the hardest college has to offer.