What is the most important thing for learning advanced calculus/real analysis?

What is the most important thing? Do you have advice for learning? Please leave any suggestions below:)
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Thank you:)

[douglasstrother]

"The Bright Side of Mathematics" has a great video series on Real Analysis. The definitions contain subtlties, but it's pretty straightforward.
Make *diagrams* of the definitions, theorems, and proof constructions. We short-change ourselves by omitting our sense of sight when studying Math, or Physics, etc.

[martinhawrylkiewicz]

Having good foundations in logic, set theory and proof writing. (Book of Proof or How to Prove it)

[KMMOS]

Through planning and scheduling do what you are able to reduce time pressure during the actual learning process. During the taking of a math course is not the best time to learn the math material. Taking the course is the time to demonstrate and show off one's knowledge of the material that one has learned before taking the class. Counterintuitive though it may seem, autodidactic efforts are when most math is learned, and accomplishing that before a class begins will allow the class to be both informative and pleasurable.

[bandman]

Cauchy sequences involve lots of thinking on the couch.

[mysticpotato]

My strategy is to put it off until my last year lol.

[peamutbubber]

Real analysis is the most fun, the best tip I have is focus on it above all else. Buy a good book, I have apostols mathematical analysis and rudin. Work on it every day, a few minutes some days, an hour or two on others, whatever you have time for. Its much more important to understand analysis first before going into other areas of maths, for me, analysis was exactly what my younger self would have wanted to learn at school instead of mindless memorising.

[josephgrossenbacher]

if 'you' really wanna learn maths then you've to have ALWAYS paper & pen at hand
when 'you' read a textbook for yourself on no matter what ( graduate ) maths-subject.
& accordingly, 'you' thoroughly use these writing tools . as a rule one might say :
per 1 h reading there's 3 h writing, then go through all 'your' writings again, 'expand' it.
good luck, though actually no luck is needed if 'you' do it this way .

[claytonbenignus]

I have concluded that the Math Major is a Bookworm's Major. When I went to High School, there were not more that two book in the whole library where I could study Advanced Math on the side. Besides, English Teachers would want you to read a book a week, so there wasn't really time for any technical stuff. Even if you got through a 100 page book on Topology or Graph Theory, most English Teachers wouldn't accept it; forget the fact you spent a month on it. The T.C. MITS series of the 40s is really Children's Literature, What do you recommend to a High School Student who wants to get a jump on Math Major Math? What yare your 10-20 favorite books for this purpose?

[davidbrisbane]

The most important thing to remember is that we all die in the end and so no matter how much or little calculus we know in the end counts for nothing, so just enjoy the ride and don't worry too much if enough calculus is enough or not.

[sklux]

Analysis happens very late in the academic journey. It is quite different from regular math. It uses completely different language and it has different approach to math-thinking in a very fundamental sense. It is about process of learning to develop maths using all the prior knowledge and understanding one gains in math studies over the years through prior levels.

Analysis takes some getting used to. Once past that stage then it gets more familiar and easier, yet never very easy. Going back and forth frequently between analysis and earlier math concepts is much helpful initially. It is as much about learning the rigour as it is about grasping the concepts. Analysis is more about devicing and planning solutions and less about solving the problems. Analysis demands more disciplined and consistent approach. Analysis is about know why/how and prior levels are about know what/how.

Introducing the rudiments of Logic and Analysis early might come handy for many who want to pursue math in higher level.

Perplexity (free AI app) is quite useful for looking up and obtaining the desired explanation/insights quickly. Main advantage is that each anwer comes with set of more questions that lead to better comprehension of the topic. One can also open up the sources of answer and verify the authenticity.

[millania]

At my university (in Germany, different system I suppose) real analysis is the first course you take and I'm actually taking it right now, eight week crash course in the holidays, 25h work/homework + everything else on top. It's devastating. I'm sitting here trying to solve questions about continuity as topological property and even though I understood everything in the lecture finally, all those creative approaches to prove writing are sooo new to me (at least I was a computer science undergraduate for one year so I've done some maths). I actually doubt my abilities cause how should I manage to get a degree if I already struggle here? It's so reassuring to here that it's the hardest course in your opinion! Everyone tells me to give up cause I'm from a not so well off background and should just go working instead of studying mathematics. You're the only person giving me motivation every since I've made that decision. Thank you so much!

[malavoy]

Most important thing is just pick up the book and get started. The book does no good just sitting on the shelf unused.

[austincarter]

Go backwards before you go forward. If you can’t lift something at the gym you’re wasting your time if you keep trying to pick up the bar. It doesn’t mean you can’t ever pick up the bar, it just means you need to build up to it. It’s odd how obvious this might seem when related to exercise, but how we completely ignore our weaknesses when it comes to something related to study. Going back also shows you it’s not because you’re incapable, it’s that there was something you missed along the way and finding that piece rebuilds confidence

[walter]

I have a some what different answer. Your mind set is the most important, Math had come easily to me before real analysis, and i didn't know what to do when i started struggling. I lost my confidence for a whie after that. You have to understand that the expectation is that you'll struggle and that you need to understand that you're probably going to need to work diferently than you did before, that your expectations about how hard you have to study need to change. You also can;t be affraid to fial and invest time and not get as much of a return on it as you used to.

[a.i.]

Can you do a video on the best math learning apps?

[jammasound]

Knew you were gonna say "Learn to write proofs"!

[yukfaicheung]

divorce automated logic proving from real analysis,
from greek's failed attempt in infinite analysis to now ...
(not the abstraction layer like modern approach).
After that, put the whole thing back in automated logic proving.

[anniesizemore]

I thought he was supposed to be going on a trip across America?

[victoronaeko]

Lol and in Nigeria we did Real analysis in our 2nd yr

[dtoor]

Hey math sorcerer, i've asked a few times and emailed you but, i'm someone who already got to calculus 1, and i want to self study the undergrad math sequence. I'm a full time software engineer. how long will it take me to get through it? (i also used to tutor math when i was younger)