TITLE: Prove that the set of integers are infinite
QUESTION [5 upvotes]: How can you prove that the set of integers are infinite? 
And can the proof be generalized to prove the set of natural numbers, rational numbers, and complex numbers are infinite?

REPLY [2 votes]: Suppose there were only finitely many integers.  Take the largest such integer, and add one to it.
To be more precise, this follows from the Peano Axioms regarding the successor function.  Namely that it is injective, that it is a natural number, that no natural number has $0$ as a successor, and lastly that $0$ is a natural number.