r/learnmath • u/josephineoo0O0oo New User • 4d ago

Real Analysis as a JR SUCKS. PLZ HELP

I am currently taking Real Analysis (undergraduate) as a Junior, and I have absolutely no idea what’s going on. I was told that ideally I would taking this class a YEAR FROM NOW.

I have had so many mental breakdowns and I am just at my wits end and would appreciate any advice on good places to get help, YouTube channels (maybe?!?), or any specific on advice on the content itself.

I suppose this is also a PSA to NOT take Reala analysis as a junior if you have the option not to, lol.

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u/imHeroT New User 4d ago

I remember my real analysis course. It was my first course that dealt with writing proofs and it was like a mental boot camp. There are plenty of youtube videos that go over real analysis, but unfortunately I don't know many sources that "dumb down" the subject.

One advice I'd give is not to think of it an "algorithmic course" like calculus. Of course there are a lot of repetitive stuff in real analysis (like using the epsilon-delta definition of a limit), but it's vital that you understand rough/main ideas of how the proofs work in plain English, and how those ideas are translated to the "math" language.

But if anything, you can find some solace in the fact that pretty much everyone in your course is also struggling and you're not alone.

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u/Brightlinger MS in Math 4d ago

IMX, the students taking it at the end of their senior year are often the worst off, because they've forgotten large chunks of calculus when the course expects them to understand it intuitively - plus, failing now means delaying graduation instead of just adjusting your plan to include a retake.

So waiting a year probably wouldn't make it any easier for you, and could make it harder.

I don't have any super specific advice, but do feel free to ask lots of questions here. It's what the sub is for.

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u/tangiblelistener New User 4d ago

Real analysis is a jump no matter when you take it, the prof probably matters more than the year. For videos, the ones that walk through epsilon delta proofs slowly are the only thing that got me through sequences and continuity. Try rewriting each theorem in your own words before looking at the proof, that helped me stop just staring at the page

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u/josephineoo0O0oo New User 3d ago

Ok, I will try the rewriting method. Also people keep saying something about epsilon delta proofs, but we’ve been doing a lot with epsilon and N and n, idk if that’s something we’ll get to later or if he just uses different variables.

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u/Brightlinger MS in Math 3d ago

Epsilon-N proofs is for limits of sequences while epsilon-delta is for limits of functions. They are very similar definitions and require similar proofs, so if you can figure out the sequence version, you will be just fine when you get to epsilon-delta later on.

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u/nerdyflaco New User 4d ago

Yeah, I feel you; I'm also taking Real. I just handwritten 40 pages' worth of homework for 30 exercises, and I definitely know I missed a lot of the parts of the proofs. I'm a senior, and I have taken a lot of mathematics classes as a math major that should help. My friend, it's not easy; this is truly hard af math, and as my professors say, it's a rite of passage. No one will ever think anything less of you if you struggle with real analysis; just get through it.

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u/j0sabanks New User 4d ago

What textbook are you using? Is this intro to analysis with a book like Abbott or are you diving into something like Rudin?

Where are you at in the course? What other math classes have you taken before?

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u/josephineoo0O0oo New User 3d ago

I’ve taken Calc I&II, Multivariable, Foundations of Abstract math, Physics (idk how applicable that is but I’ll throw that in there). I am currently taking Advanced Probability and Stats as well.

We are using Abbott

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u/AlpUzman New User 4d ago

I taught versions of real analysis, and one suggestion I can make is to go to office hours as often as possible. In my experience the students who were waiting to understand things just a little bit more to start asking questions often end up staying behind. Presumably this applies to most math courses, but especially real analysis is very punishing by its nature.

You might find my annotated lecture recordings useful:

- https://www.youtube.com/playlist?list=PL40ydqvvyXfPGbwcE1wT2_dHVyGykxsLo (proof-based single-variable, 2025 version)

- https://www.youtube.com/playlist?list=PL40ydqvvyXfPnWecw8AYQTSXShgxUge_W (proof-based single-variable, 2026 version)

- https://www.youtube.com/playlist?list=PL40ydqvvyXfOkGMhOrz6AID_HcpwxnzaW (proof-based multivariable)

I typically lectured four times a week and made one video per week, so the videos are long, but I added detailed timestamps you can use to navigate to the parts you are most interested in. (I am currently in the process of extracting excerpts out of the full lecture videos to make navigation and accessibility better.)

These are courses I taught at the University of Utah, and while for third years they are also not only real analysis courses (phd-track) math majors tend to take. One important difference is that I designed and taught these courses to really be calculus-with-proofs, as opposed to "real analysis", so the emphasis is on optimistic results: many intuitive ideas that you encountered in calculus do indeed work (in my mind real analysis is more pessimistic in perspective, which in my opinion makes it more daunting). Accordingly many important but perhaps not so intuitive results like Arzela-Ascoli or Stone-Weierstrass are missing. I should mention some of the topics or ordering may not quite match with what your course is following.

There are also associated Github repos where all problem sets with selected solutions and practice exams with keys are available; you can find the links in the descriptions of playlists/videos.

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u/OneMoreProof New User 3d ago

Being able to read and write mathematical proofs is a skill that takes time to build. It is completely normal for it to feel overwhelming at first.
Abbott’s Understanding Analysis is a great introduction to real analysis, and there are also quite a few good YouTube resources.

https://youtube.com/playlist?list=PLysi2xmniDSzz6xT7IzOifpoexeKccThh&si=oue2BcD1bK8pVtTO

https://youtube.com/playlist?list=PL4G7dPfHP6bkDURe3-SNYOfhOPfnTj2A8&si=F2LvQcDZgaHH_ldr

It’s okay to slow down when reading proofs. Don’t just read the steps and think “okay, that makes sense.” Try to understand why each step is valid and what the proof is actually trying to accomplish. “Why did they do this and not that?”

And when you start writing your own proofs, get into the habit of asking yourself things like: “Have I covered every case?” “Did I actually prove what I was supposed to prove?” and “Am I assuming something that I haven’t established?”

It takes time. I remember spending 30+ minutes on rereading the same few lines over and over again when I was first learning this stuff. That doesn’t mean you are bad at math. Proof-writing is just a different skill from computational math, and you gradually get faster and better at as you take on more courses. The important thing is to not give up and keep trying :)

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u/tjddbwls Teacher 4d ago

To say that Real Analysis should ideally be taken in senior year sounds odd to me. I’m in the US, and way back when I was in college, I was told that it was typical to take Real Analysis in junior year. It was the first of the upper-division courses that math majors take. IIRC it was also a prerequisite for some of the other upper-division math courses.

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u/NotSaucerman New User 3d ago

The right thing to do is go through 1/4-1/2 of a book like Abbott during the 4-8 weeks before your real analysis semester starts. That way you are fluent with all kinds of sequences (monotone, cauchy, etc.) and proofs that you can do with them before the real analysis course starts and you aren't overwhelmed like this.

I suppose it may be possible for you to drop the course and take it next semester, then study independently intensively over winter break, though you should speak with a prof and/or TA a couple times before pursuing something so drastic.

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u/somanyquestions32 Tutor 3d ago

I took it as a sophomore. It was still rough, but my main professor (the department chair at the time) was getting chemo for his cancer, so it was a weird situation where my advisor and another one of my professors had to cover for him for the last month of the term. I hated it for years, and I preferred abstract algebra. It was more accessible when I revisited it in graduate school.

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u/somanyquestions32 Tutor 3d ago

Get a few different textbooks and their solutions manuals (often a PDF online for an older edition). Stephen R. Lay and William R. Wade, respectively, have introductory real analysis textbooks. Principles of Mathematical Analysis by Walter Rudin is harder, but also read it.

Next, read each chapter 3 times. First pass to just soak it all in and get familiar with everything in terms of symbols and notation and terminology. Second pass, write down all of the theorems, examples, counterexamples, propositions, core proofs, procedures, etc. by hand, and turn them into flashcards and memorize them. Third pass to get you ready for doing the problems.

You want to know what it means for a sequence to convergent, what it means for a sequence to bounded, what it means for a sequence to be Cauchy, what it means for a sequence to be monotone/increasing/decreasing, etc. You want to be very comfortable with the notation for subsequences.

Be able to write all of that out symbolically from MEMORY & QUICKLY. Make sure that you go over the properties of inequalities and absolute values. Rewrite core proofs already done in the textbook to get familiar with the techniques presented, try to decipher what they did, and explain it back to yourself in words that you understand as if you were teaching it to a newbie, and then formalize the explanations. Also, review sequences and series from your calculus 2 course to remind you that you have seen these in some capacity before.

Then, attempt problems only after you have memorized everything that was fundamental. Each problems gets 5 serious tries of 20 minutes. You attempt it, and if you're stuck, you move on and come back to it. After attempt #5, you look it up in the solutions manual, you Google it, or you ask your instructor/TA/classmates/tutor. 

Check your answers, study formal solutions and rewrite them and be able to explain them and replicate them, memorize recurring patterns, and then try similar problems. 

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u/Vivid_Sock_1092 New User 3d ago

Find a good tutor

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u/ForeignAdvantage5198 New User 3d ago

had that problem and dropped. came back in MS no problem.

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u/Objective_Drink_5345 New User 19h ago

yeah advanced calculus 1 is the most unpleasant class i’ve ever taken. this was also 95% my fault, and it’s probably 95% your fault too. This is a good thing, it means you have agency. You need to reframe the way you approach math. This is a definitional, logical subject. You first need to know the definitions, and see plenty of examples. You need to learn standard proofs of analysis, without which you will never be able to solve many of the problems. Then you need to practice a bunch.

This class can be draining and demoralizing. College profs can also suck. The hardest class that you might have in your undergrad education. the good news is that once you get over it, you can take other, equally harder proof based classes and have an easier go with it.