Infinitude of Primes

For an alternative to Euclid’s proof, there is Furstenberg’s topological proof of the infinitude of primes. The MAA‘s Mathematics Magazine published a variation of Furstenberg’s proof which does not use topology:

For each prime p, let Sp = pZ. Each Sp is periodic (it’s characteristic function is periodic). Let S be the union of all Sp. If S is the union of finitely many periodic sets, then S is also periodic. However, the complement of S is {-1, 1}, so S is not periodic. Hence, there must be infinitely many sets Sp, and infinitely many primes.