Natural Numbers are Infinite

Theorem
The set $\N$ of natural numbers is infinite.

Proof
Let the mapping $s: \N \to \N$ be defined as:
 * $\forall n \in \N: s \left({n}\right) = n + 1$

$s$ is clearly an injection. But:
 * $\forall n \in \N: s \left({n}\right) \ge 0 + 1 > 0$

So $0 \notin s \left({\N}\right)$, and $s$ is not a surjection.

Therefore $\N$ is not finite and so by Same Cardinality Bijective Injective Surjective is therefore infinite.