[Math] Proving $f:\mathbb N \to \mathbb N; f(n) = n+1$ is not surjective.

Question

Prove that the function $$f:\mathbb N \to \mathbb N; f(n) = n+1$$ is not bijective.

So I know that we can prove it is injective because we can suppose or let $n_1$ and $n_2$ are natural numbers with $f(n_1)=f(n_2)$, then by definition of f, that means $n_1+1=n_2+1.$ When subtracting 1 from both sides we know that $n_1=n_2$.

How do I prove it is not surjective? I know that for any natural number we put into the function that it would just be plus 1, but how do we prove that? It is one-to-one.

EDIT: So I think this proves it.
$$f(0) = -1$$
So -1 is not in the natural numbers set, how do I correctly write this for a proof?

Answer 1

Hint: What is the least element of $\Bbb N$? (Can you see why that's relevant?)