Introductory material before real analysis

Question

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?

Answer 1

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. =)

Answer 2

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!