1

I'm learning real analysis at the moment, however, I'm having a lot of difficulties. Although I have already learned univariate and multivariate calculus as well as analytical geometry and linear algebra I can rarely prove anything in real analysis, moreover I have acquired a bad habit of transcribing demonstrations rather than doing them.

In general I can understand the demonstrations when I see them, however when I try to do it myself I can not.

Any material recommendations before I go back to face real analysis?

Roland
  • 3,165
  • Look at similar question at https://math.stackexchange.com/questions/63732/how-to-study-for-analysis – CopyPasteIt May 30 '17 at 00:58
  • Since you are weak in proving stuff, how about this for a summer project: construct the real numbers from the rational numbers on your own. c,f. https://math.stackexchange.com/a/1217816/432081 – CopyPasteIt May 30 '17 at 01:26
  • 1
    If using an easier book, such as Ross's Elementary Analysis doesn't work (or whatever an equivalent book in Portuguese would be), you may find that the best way to improve is to work with a teacher one-on-one. I don't know what it's like in Brazil, but often in the U.S., university math departments have lists of PhD students who also work as tutors. – user49640 May 30 '17 at 02:07
  • Given that the vast majority of modern technical material is written in english (bar a few fields), it wouldn't be a bad idea to learn from a textbook in english. – Batman May 30 '17 at 03:03
  • @user49640 I have this Ross's book I will take a look at it. – Roland May 30 '17 at 03:10
  • @Batman I will leave it up to Roland to decide what language he wants to read math books in, and I have no reason to believe that there are no books in Portuguese that would be better than the one in English that I'm suggesting. I was simply pointing out that answers he gets here are likely to be biased towards English books, so it might also be worthwhile to ask Portuguese-speakers what similar books are available in that language. – user49640 May 30 '17 at 03:23
  • "In general I can understand the demonstrations when I see them, however when I try to do it myself I can not." @Roland, sounds like you need to study a bit of logic and reasoning. – shredalert Jun 04 '17 at 09:02
  • @shredalert What material you suggest? – Roland Jun 04 '17 at 12:14
  • @Roland some study of deduction systems will help you prove things on your own. – shredalert Jun 08 '17 at 11:10

2 Answers2

3

Depending on what real analysis book you're using, it may help to use an easy one as a supplement, like "Elementary Analysis: The theory of calculus" by Ross. Marsden's "Elementary Classical Analysis" also has a lot of worked out material, which may be helpful next to some of the more common texts like Rudin's Principles of Mathematical Analysis.

Also, a book that has an introduction to proofs, like Vellman's "How to Prove it", or West + D'Angelo's "Mathematical Thinking: Problem-solving and Proofs" may be useful as well.

At the end of the day though, it's likely a question of effort and mathematical maturity, which means being exposed to the material for a while and suffering working through it. =)

Batman
  • 19,390
  • But is bad to learn if I look at the proofs of theorems without try it myself? – Roland May 30 '17 at 00:20
  • Its nice if you can prove theorems before looking at the proofs. But, at the minimum, you should understand the proofs thoroughly. – Batman May 30 '17 at 00:25
1

Since I'm not a native English speaker, I can't really give you a good material recommendation apart from some classic books (which you probably already know of and people will probably suggest anyway). While not really answering your question, my suggestion is too long for a mere comment.

First, get a highlighter pen and mark the definitions, theorems, propositions and such. Not the demonstrations, but the statements. Some demonstrations only combines them in a meaningful way.

After that, carefully read the demonstrations and note down the overall steps. Don't detail them too much, only the main tricks and techniques.

Next, try to demontrate yourself some theorems with your own words. Try not to look into the guide you made, but feel free to review the previous theorems you highlighted. If you really hit a wall, then only read the next step needed.

Then, go for the exercises. Remember the most used techniques you read before, because they're likely to be used in the exercises too. Some exercises may also follow the overall proofs in the text, but adding or removing conditions.

Try also to find people that are interested in studying with you, if possible. Last but not least, when you get really stuck, don't hesitate to ask us!

AspiringMathematician
  • 2,218
  • 12
  • 28
  • I am not a native english too, I'm using "Análise Real, Elon Lages Lima", but find it a litle hard. I'm not a mathematician but a statistician – Roland May 30 '17 at 00:49
  • 1
    Nice to meet a fellow Brazilian here too :) I had it rougher with his "Curso de Análise vol. 1". Analysis can be quite dense if you see it for the first time, but don't let it drag you down. – AspiringMathematician May 30 '17 at 01:05
  • 1
    And adding up, some search on Google mention Apostol's "Mathematical Analysis" and Terry Tao's "Analysis I". Maybe you'd want to take a look at it if you haven't yet. – AspiringMathematician May 30 '17 at 01:07