Bijection/Examples/n+1 Mapping on Integers

Example of Bijection
Let $f: \Z \to \Z$ be the mapping defined on the set of integers as:
 * $\forall n \in \Z: \map f n = n + 1$

Then $f$ is a bijection.

Proof
Let $n_1$ and $n_2$ be integers.

Then:

Hence by definition $f$ is an injection.

Let $m \in \Z$.

Let $n = m - 1$.

We have that:
 * $n \in \Z$

and:
 * $\map f n = m$

Hence by definition $f$ is a surjection.

Thus $f$ is both an injection and a surjection, and so a bijection