What are some good proofs to read? I have just started my second year in a maths degrees and I am interested in reading mathematical proofs, I find the proofs to everything I do in class fascinating so I'm looking for some proofs to some more difficult problems that I can try and wrap my head around. I obviously can't jump into something like an attempt at proving Fermat's last theorem, I'm just looking for something to try and challenge myself to make sense of.
 A: The way to understand a theorem, and what a theorem does, is to weaken the hypotheses and find examples that fit the weakened conditions but no longer quite fit the theorem conclusion. In brief, examples. See if I can find the Kendig book quote... 


A: *

*Erdos' probabilistic method is really cool in that it uses ideas from probability in a very unique and clever way.  There's some background reading you'll need to do, but not much.  At the risk of self-promotion, while we're in the Ramsey theory area, check out the proof for $R(3,3) = 6$.  A neat little idea, easy to digest even for a (precocious) elementary or middle school student, and yet a satisfying nugget of creativity involved.

*Another favorite of mine (due to its application to a potential magic trick for a skilled card-shuffler) is figuring out how many "perfect" shuffles are required on a deck of $n$ cards to return the stack to the original ordering (way fewer than you might think for some $n$).  A little group theory knowledge is required here, though.  

*There are some accessible proofs of some neat facts that involve little more than a clever application of the pidgeonhole principle.  Googling would almost certainly turn something up in this regard.  Ah, yes.  Here's one:  Consider the points of $\mathbb{Z} \times \mathbb{Z}$ in the plane.  Color each either red or blue.  Prove that we can find a rectangle with monochromatic vertices.

*The compass-and-straightedge construction of a pentagon: more of a proof-by-demonstration, but thought-provoking nonetheless.

*Set theory: prove that there are infinite sets of different cardinality (this one is really important).  In particular, $|\mathbb{N}| < | \mathbb{R} |$.

*More set theory: prove that, when $S$ is an infinite set, $|S| = |S \times S|$.

*Even more set theory (the easiest of the three): Prove that a set $S$ is infinite $\iff$ there exists a subset $T \subset S$ such that $|S| = |T|$.  As a side note on this one: it was a magical, eye-opening experience for me seeing for the first time that the cardinalities of the Cantor set and of $\mathbb{R}$ are equal.

*The existence of asymmetric cryptosystems.  This was another shock-and-awe, "I cannot believe this is possible" moment for me.  The premise is that two strangers who've never met can call each other on the telephone and, using some mathematical trickery, manage to exchange secrets between each other even if the telephones are wiretapped by an opposing party.  Read about Diffie-Hellman in particular; it's fairly simple but requires some basic group theory.
I'll append more to this post as things come to mind.
A: Based on what you've said I'm inferring you've had minimal experience reading and (more importantly) writing your own proofs. Although reading proofs of specific theorems can be enlightening, I think that a more systemic approach to learning about proof based mathematics is preferred to reading random proofs.
How to Prove It: A Structured Approach by Daniel J. Velleman is a bit of an eclectic book, but a good introduction to proofs. It might be the best first book to read, especially if you have minimal experience writing your own proofs.
Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand, Polimeni, and Zhang a good grounding in proofs, taking time to developing a variety of proof techniques and set theory. Understanding both of these things (at least a little) is very important in mathematics.
Once you've read these two books, I think that you'd be ready to start reading and writing your own proofs, at least basic ones. Check out these problems by Professor Laci Babai. They have been a continuous gift in terms of fun things to think about. Most of them have proofs that have an "ah-ha" moment, and the core concept can be communicate in a single sentence (or even word). Another advantage of these problems is that they don't look like mathematics problems, and a fundamentally important skill is to learn to think mathematically about things that don't look like mathematics problems. Another great resource is this website. Obviously the quality is massively varied, but go read high rating answers, or answers from high rated users. Or just whatever strikes your fancy tbh, the average quality here is pleasantly high. If you want to learn about a specific subject at this juncture, these two books come to mind as great first books in their fields:
Calculus by Spivak is probably the best calculus book if you haven't done proof based calculus. In my mind this book is just much better than other similar books.
An Introduction to the Theory of Numbers by Hardy and Wright would be a pretty good guide to an introduction to number theory. You might have to consult online references or other books for some context, but it's one of the best number theory books out there. Number Theory is an often ignored but fundamentally important subject, both to modern mathematics and to the history of mathematics. I very much enjoyed watching my friends taking number theory their third or fourth year gawk at the myriad connections to other fields.
A: I would recommend something in elementary number theory (which covers areas like primes and divisibility). It is a very accessible field, and many of the early theorems in it are amenable to testing by hand.
Divisibility tests, such as the process of "casting out nines" for example is taking advantage of modular arithmetic, part of number theory. E.g. $8451$ is divisible by $9$ since $8+4+5+1=18$, and $18$ is divisible by $9$.
I personally like "The Theory of Numbers" by Niven and Zuckerman, but there are probably newer texts that also well suited for a beginner.
A: You can try the wonderful book 'Proofs from THE BOOK' by Martin Aigner and Gunter Ziegler. 
This book is a collection of mathematical proofs where you have more than one proof for a theorem. This book was dedicated to Paul Erdös! 
If you want to get a glimpse as to how the greatest minds in mathematics worked, try the book 'Journey through genius' by William Dunham. 
A few books about problem solving in mathematics may help you understand how to prove 
1)The Art of Mathematics by Bollabas 
2)Berkeley Problems in Mathematics by Silva
3)1001 problems in classical number theory by Konnick
