Question

prove that the set of natural numbers is infinite.

Answer 1

Let the mapping s:N→N be defined as:
∀n∈N:s(n)=n+1
s is clearly an injection. But:
∀n∈N:s(n)≥0+1>0
So 0∉s(N), and s is not a surjection.
Therefore N is not finite and so by Same Cardinality Bijective Injective Surjective is therefore infinite.
http://www.proofwiki.org/wiki/Natural_Numbers_are_Infinite