Analysis I

What Hacker News says

1. hamburgererrorJul 2026
   Thanks for the references, I'll have a look. I can tell you what I don't want: books that throw theorems one after the other without any context, like Baby Rudin [1] for instance.

   I tried once to read Terence Tao's Analysis I [2], it's really good but my main problem was that I wasn't able to know if my proofs are correct when I do an exercise. So maybe the solution is to get a teacher.

   [1]: https://en.wikipedia.org/wiki/Principles_of_Mathematical_Ana...

   [2]: https://terrytao.wordpress.com/books/analysis-i/

   Show morethread
2. MetroWindJun 2022
   I'd recommend starting from Terence Tao's Analysis I book. It really starts you from "zero" (literally), and constructing natural numbers, all the way to real numbers and integrals, and beyond.
   thread
3. sunny3Feb 2022
   Introduction to Calculus and Analysis, by Richard Courant and Fritz John. This book clearly explained many of the motivating examples of the concepts in Calculus, which was important to me as a beginning student. Too many times I read math textbooks that simply present a list of definitions, theorems, corollaries, and whatnot, failing to guide readers from one point to another. I agree that for advanced level of math, there're probably no good motivations or 'storyline' that weave these constructs together (they're purely symbolic); however, for a layman level, I feel it's crucial to at least make the readers understand why and how these concepts come to be, so that they can grasp at least a little bit of the 'logic' of math.

   Similarly, Analysis I by Terence Tao has a 'Why do analysis' that has made me actually enjoy analysis.

   Show morethread
4. cevinkJun 2021
   You start by writing your own examples. Remove all the abstract notation and expand out everything, distribute everything back in so you just have a series to be summed or integrated. Stare at it for k = 0, 1, 2, etc., like 5 choose 4, 5 choose 3...maybe you can prove this yourself. This is actually how you learn, trying to do it yourself and thinking for a few hours/days/weeks about this problem why the identity is true, and then getting all those 'How to prove it' style books filled with established techniques to see if your identity can be proved by them.

   Try Terence Tao's book Analysis I https://learnaifromscratch.github.io/math.html you pick up the gist of this kind of math doing basic proofs of natural numbers. You see and do enough of these that you will get the first proof here, where the n+1 comes from, and how you can distribute things in/out of any summation notation. The second integral proof from math stackexchange the same applies, remove all the abstract notation and just write down a few series, distribute in the 1/2pi(i). Do this for a few examples and you can probably figure it out yourself.

   In the above link there is resources for olympiad and putnam problem solving on YouTube, if you watch some of those you'll see how they train competitors to solve something like this problem. List every tiny idea or strategy you have to prove that binomial identity and pick one that seems the easiest, start there. Fill the variables with corner cases what happens. If it's geometry what happens if you 'stretch and pull' the geometric interpretation by some constant factor. Look at Pascals Array since this is a binomial identity. This is what they do in those seminars if you're interested and it's free for anybody to watch.

   When all those proof books throw 'implies' and other logic at you, read through this: https://ncatlab.org/nlab/files/MartinLofOnTheMeaning96.pdf

   Show morethread
5. hackermailmanApr 2020
   Try Terence Tao's book Analysis I, start at chapter 2 if you forgot trig. Defines naturals, integers, rationals and reals from scratch teaching you how a proof works at the same time. When you get to the calculus chapters watch Gilbert Strang videos with the book if you want extra insight but Tao's book is clear enough without needing secondary resources. There's a free sample of the first few chapters and the excellent appendix on basics of math logic https://math.unm.edu/~crisp/courses/math401/tao.pdf

   All those proofs in probability and stats/linear algebra classes will make sense after imo

   Show morethread
6. hackermailmanNov 2019
   at 5:30 Wildberger goes through the definition of Limits https://youtu.be/K4eAyn-oK4M?t=331

   If you really want to understand these things, start here: https://www.youtube.com/playlist?list=PL5A714C94D40392AB and watch them as you work through Analysis I by Terence Tao, skipping to chapter 2 where he begins from scratch introducing the naturals, integers, etc. Wildberger even shows you how to do algebraic calculus which will help to learn the Tao book when he goes about rigorously introducing calculus defined using limits. For me that helped anyway, seeing Naturals, Ints, Rationals, and Reals constructed from scratch with all their laws/operations defined and proved by Tao then him using analysis to introduce how these tiny changes in calculus work.

   Show morethread