Natural Numbers are Infinite

Theorem
The set $$\mathbb{N}$$ is infinite.

Proof
Let the mapping $$s: \mathbb{N} \to \mathbb{N}$$ be defined as:

$$\forall n \in \mathbb{N}: s \left({n}\right) = n + 1$$

$$s$$ is clearly an injection. But:

$$\forall n \in \mathbb{N}: s \left({n}\right) \ge 0 + 1 > 0$$

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

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