I have just recently completed a course in a basic proofs/logic and will be transitioning towards Rudin's "Principles of Analysis". Despite having some ground in logic/proofs, I find that the proofs in Rudin's book is very clever and concise and short, which is a no brainer I guess since it is a classic.

For example, at first I was struggling to understand the Definition-Theorem style of the textbook without much commentary but after following the lectures on Real analysis by Francis Su, I find that I am beginning to understand things and its not as complicated. However, the next issue is the exercises since the proof/logic course I took taught me to be as concise as possible.

So doing Rudin's first problem of: Let $r \in \mathbb{Q}, x \in \mathbb{I}$, prove that $r + x \in \mathbb{I}$ and $rx \in \mathbb{I}$.

The routine I did was a proof by contradiction using cases first showing that $r + x \in \mathbb{I}$ by letting $r = \frac{a}{b}$ where $a,b \in \mathbb{Z}, b\neq 0$. $r + x = \frac{p}{q}$ where $p,q \in \mathbb{Z}, q \neq 0$. Then showing that $x = \frac{pb - aq}{qb}$ where $pb-aq, qb \in \mathbb{Z}, qb \neq 0$. Which leads to a contradiction and similarly fpr $rx \in \mathbb{I}$ as well.

However, the solutions to this exercise is a 2 line proof showing that $x = r + x - r$ would also be rational for the first case.

So my question, which I'm sure is common to many beginners is how would one recommend making the transition towards doing / understanding the level of proofs done by Rudin, which is very concise and to the point. I'm sure this would help many beginners starting out with Rudin and even reading to the level of proofs done in mathematical papers.

Advice would be greatly appreciated.


You are very correct in saying that this is a common point of concern for people getting into proof-based mathematics. Here are a few things that either I was told or did that helped me:

1) It takes time. Training your brain to think "backwards" can be a difficult process as you have experienced and everyone makes this transition at a different rate. Working more proofs will probably help speed this up.

2) Work proofs. I have caught myself many times reading mathematics like I would a good novel and that is a dangerous thing. I believe a good practice would be to read the statement of the theorem, try scratching out a proof yourself, then read Rudin's. Look for the similarities, and try to gain insight as to why those specific versions of the proof made it into the book.

3) Look for other versions of a proof if you cannot understand or do a proof yourself. Rudin's book is great (I'm working out of it myself) but it is not the only resource available. Sometimes even going and finding a less concise or clever proof and understanding it can allow you to understand the more concise version.

4) I don't know what your proof course covered but something my old number theory professor told us to do was write out everything we know (hypotheses and their implications), and what you are trying to show. The proof is the road map between those two sets of things.

5) Have fun with it, find friends to work with and talk about proofs and their method of attack on problems. Talking through a math problem is an extremely powerful and often enjoyable process.

  • $\begingroup$ You cannot become an expert by reading one book. $\endgroup$ – Rene Schipperus Aug 28 '16 at 2:17

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