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I am a recent graduate with mathematics degree but I still don't think I truly understood all those difficult courses, especially the real analysis. My professor told me during his lecture that nobody really understands this stuff, they just get used to it. I was feeling frustrated because I seemed to be studying mostly using my memory rather than truly understanding the theorems. I am studying over the book I used for my introduction to real analysis because I really want to feel like I learned what real analysis is.

Now, it is easier for me to grasp the idea of proofs but still I don't think I will ever come up with these proofs on my own. I am doing the proof analysis which is basically re-writing the proofs in my own words. But would this be a really efficient way of studying real analysis? Should I keep trying to prove the theorems on my own even if might take me a whole day, or even a week? Or should I try to come up with my own proof? I feel like I slacked through the classes without really doing mathematics even though I got good grades in the end. Sometimes I feel like I don't have any talent in doing mathematics so why I should even bother trying. Please give me advices on how I can more effectively study and learn mathematics of upper level.

user3000482
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    Regarding real analysis specifically: I personally found that a lot of real analysis made a lot more sense once I took a course in topology. That being said, the intuition I had from analysis made taking topology much easier. Keep in mind also that it's very easy to take the things you already know for granted; it's very easy to feel like you haven't learned anything after you've spent the semester learning just enough to know how much you don't know. – Ben Grossmann May 24 '17 at 03:38
  • Have you tried exercises in real analysis? Instead of trying to prove one of the big theorems whose proofs are carefully given to you in the textbook, try to answer some exercise questions. By design, they challenge you to construct a proof from scratch, but the proof shouldn't be as involved as it might be for one of the bigger theorems. – Ben Grossmann May 24 '17 at 03:41
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    Trying to come up with proofs can be very time-consuming, especially if you are working through an advanced textbook, and it might even hinder your learning. Give the theorem some thought, stare into the wall for like 5 minutes, then read the proof. Understand it, then write it down in your own words. Work through a lot of exercises, and whenever needed, memorize certain techniques. If a theorem comes across as a bit overwhelming, don't get discouraged and keep reading and scribbling on paper until you understand it. –  May 24 '17 at 04:48
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    Actually, in particular with real analysis, you must focus on the counter-examples, the weird cases. Most of the stuff is the same, but the only thing giving you an alert of what you should be aware is the weird cases. Those with infinite oscillations close to zero, the converging/diverging, etc. Some teacher told me maths is a set of counter-examples. – Brethlosze May 24 '17 at 05:14

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I think the real obstacles in understanding any subject are 1)lack of good books, 2)lack of good teachers and 3)lack of interest. You have a choice in fixing 1) and 3).

For real analysis you must have a thorough idea of what real numbers are and how they are different from rationals. Unfortunately most textbooks skip this part and jump onto the bandwagon of analysis concepts like limits/continuity/derivatives/integrals. And you have to seriously stop thinking in terms of algebraic manipulation of symbols. Real analysis is concerned with order relations of real numbers (denseness) and not the field operations $+, -, \times, /$. Sadly most textbooks try to treat analysis also in mechanical fashion by developing mechanisms of symbol manipulation almost like in algebra.

One such technique of symbol manipulation is the use of pointless $\epsilon, \delta$ exercises and some books also go to the extremes to indicate that these exercises are mostly algebraic in nature almost equivalent to solving inequalities in order to get $\delta$ as some algebraic expression in $\epsilon$. This is so counter-intuitive to the whole spirit of analysis. Perhaps a student is much better off if he learns the $\epsilon, \delta$ techniques from the proofs of various limit laws rather than trying these pointless exercises.

Another key aspect in analysis is a very clear understanding of the completeness of real numbers (and it's expression in various forms). Without this you can not do anything in analysis. All the significant theorems of analysis are based on it. This topic is not difficult but rarely analysis textbooks focus on this aspect. Instead it is just presented as a one line axiom and then used almost everywhere in proving theorems.

I would suggest two books: A Course of Pure Mathematics by G H Hardy and Mathematical Analysis by Tom M Apostol which do not have these drawbacks and you should study them in that order. They are nowhere as popular as Rudin's books but they are much better if your goal is understanding analysis rather getting good grades in some exam.

Paramanand Singh
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