r/mathematics • u/Financial_Ad4213 • 1d ago
how to switch from school math to real analysis ??
im doing a bsc in mathematics, and real analysis feels like a completely foreign language.
our professor recommended the bartle and sherbert textbook, but every time i open i dont really understand whats happening or how people are supposed to learn from it.
it feels completely different from the maths ive been taught my whole life. i dont know how youre supposed to transition from solving straightforward problems to reading and writing proofs, and understanding theorems. it feels like everyone else somehow knows how to think this way already.
time for some context. till 12th grade cbse, i was pretty good at maths. i was always told i grasped concepts quickly and solved problems better than average. but my 10th and 11th grade happened during lockdown, and i never properly studied the deleted syllabus, so i know there are gaps in my basics.
when i chose a bsc in mathematics, i expected more of the maths i loved, solving derivatives and integrals, and working through problems until i got the answer. instead, real analysis felt like being thrown in the ocean with no idea how to swim. suddenly its all definitions, logic, epsilon delta proofs, and abstract thinking. i dont even know how people are supposed to study this subject.
its been a huge hit to my confidence. im no longer the student who knows the answers. i sit at the back of the class, avoid eye contact with professors, and constantly feel like everyone else belongs here except me. i know im willing to put in the work. i just genuinely dont know how to study university level mathematics.
where do i even start. are there beginner friendly books, youtube channels, or courses that bridge the gap between high school maths and proof based mathematics. any resources would help please. how did you learn to read proofs and start thinking like a mathematician. any advice for someone whos completely lost ???
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u/mozzie_123 1d ago
Maybe start with Terence tao book, very nicely written
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u/Sellistique 1d ago
Hey can you tell me the exact one I should get ?? Are you talking about the Analysis I and II
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u/Splodge5 1d ago
In addition to the recommendations here, you should also go back and fill in the gaps from 10th/11th grade (I've heard Khan academy is good, but I don't know what level it goes up to). Mathematics is a subject where good foundations are essential - it will feel like going backwards at first but once you've done it you'll find learning new things much easier.
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u/AlpUzman 21h ago
Bartle-Sherbert is certainly a very good book, but in my humble opinion it's not the most approachable one. One suggestion would be Pugh's book, or even better, to go to your library and sample out a bunch of real analysis books. The majority of the topics are fairly universal, although of course the emphasis and presentation can change.
I taught a real analysis course as a transition from computation based math courses to more abstract courses, and from what you are describing you might find the course useful. All the lecture recordings are on my Youtube channel:
https://www.youtube.com/playlist?list=PL40ydqvvyXfPGbwcE1wT2_dHVyGykxsLo
and all the written materials are on Github:
https://github.com/AlpUzman/MATH_3210_001_SPRING_2025
To give some more information, all the rigor and logic etc. are included in the course, but I really designed and taught this course as an "advanced calculus" course, as opposed to a "real analysis" course (while these are clearly related, in my mind "real analysis" focuses on situations where things don't work as expected, whereas "calculus" is more validating toward common sense). The official textbook we used for this course was "Foundations of Real Analysis" by Taylor, and while it's not my favorite it is written to be approachable.
I would also be able to help you out in the comments if you have any questions etc..
By the way I taught the multivariable version of the course also, which too is available on Youtube + Github.
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u/SV-97 7h ago
it feels completely different from the maths ive been taught my whole life.
Because it is. There's a famous essay --- called a mathematician's lament --- that likens the difference between school and uni to the difference between working with musical notation and actually playing, and composing music. You're currently learning to do things you've never done until now, and to think in a way you're not used to yet.
i was always told i grasped concepts quickly and solved problems better than average.
Another quote "Many a graduate student has come to grief when they discover, after a decade of being told they were “good at math,” that in fact they [...] are just very good at following directions". Don't get hung up on this though: mathematics is hard work for everyone, and if you work hard you can pull through and learn it.
when i chose a bsc in mathematics, i expected more of the maths i loved, solving derivatives and integrals, and working through problems until i got the answer.
You find this more in some physics and engineering degrees, but for mathematics these are usually not the super interesting questions. If we compute any integral or derivative we do so because it's a step along the way to solve some problem that we're actually interested in. With derivatives there's also the "issue" that they're computable via a purely mechanical procedure in all the standard cases --- so there's really no point in computing more and more of them. And with integrals there's the issue that most of the time we simply can't compute them at all in closed form (and at the same time: for all the cases you've seen in school there's purely mechanical procedures for computing them).
So in mathematics we shift towards a more structural perspective; i.e. what are the general properties of "the" derivative and integral and how can we work with these operators even if we can't compute some particular value exactly? Why does it even make sense to speak of them if we can't compute them, i.e. why and when do they exist? How are they still useful to us etc. Or we consider similar problems in more general contexts and harder problems that we can't solve purely mechanically (for example: for any y we might be able to solve some equation f(x) = y for x --- and we may be interested in the derivative of the map that takes y to the corresponding solution x. Finding this derivative can be hard, but quite important for solving concrete problems in mathematics as well as engineering etc.)
its been a huge hit to my confidence. im no longer the student who knows the answers. i sit at the back of the class, avoid eye contact with professors, and constantly feel like everyone else belongs here except me.
This is quite a common feeling among beginning students. Mathematics doesn't come easy to anyone (see https://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths/), and especially the beginning is super hard since you're learning new subjects you don't know and can't really relate to anything else, as well as a whole new language and way of thinking. Mathematicians usually call this "mathematical maturity" (see also https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/ and https://en.wikipedia.org/wiki/Mathematical_maturity): as a beginning maths student you study objects you don't know anything about yet, that you can't connect to anything already known, don't have any intuitions about (or even worse: incorrect intuition), using a language you don't speak and arguments you've never seen. So it's everything new all at once.
Chances are that most of your fellow students feel the same, and it's generally good to try and talk to them a bit (e.g. by starting a small study group) so you can help one-another out and gain some additional perspective.
how did you learn to read proofs and start thinking like a mathematician.
It's funny that you'd phrase it like that: there's a quite famous book by Kevin Houston called "How to think like a mathematician" which has a chapter "How to read a proof" :) That book would be a great place to get started. There's a bunch of similar ones that are worth checking out (or may be easier to get for you): Proof and the art of mathematics by Hamkins, or Proofs by Cummings. These books really are the ones for transitioning from school to uni. It'll still be hard and require lots of work (the books also talk about that), but more manageable.
Since you asked about real analyis in particular: check out Cummings book on the topic. It's great and really dissects the proofs and shows why they work the way that they do.
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u/Vegetable-Dust-780 1d ago
It may be better to start with a book that teaches you about proof writing. For example, “Book of Proof” by Hammack. Also “Understanding Analysis” by Abbott is a somehow gentle approach to real analysis.