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Yay! I really love Euclid proof of The infinitude of primes. It is so short, simple & elegant that any starting learner could understand it easily.

http://primes.utm.edu/notes/proofs/infinite/euclids.html

I put it here: Call the primes in our finite list p1, p2, ..., pr. Let P be any common multiple of these primes plus one (for example, P = p1p2...pr+1). Now P is either prime or it is not. If it is prime, then P is a prime that was not in our list. If P is not prime, then it is divisible by some prime, call it p. Notice p can not be any of p1, p2, ..., pr, otherwise p would divide 1, which is impossible. So this prime p is some prime that was not in our original list. Either way, the original list was incomplete.