Proof:
Suppose $X$ and $Y$ are any infinite sets.
Then
$$ \exists f:X \rightarrow X \text{ such that}\; f \;\text{ is injective }\;\land\; f(X) \neq X,\;\;\text{and}$$ $$\exists g:Y \rightarrow Y \text{ such that}\;\; g \text{ is injective}\;\;\land\;\;g(Y) \neq Y.$$
I'm sure that it's simple, but I don't see what I should do after this.