I've come to this:
$$f: \mathbb{N} \to \{\ldots, -6,-4,-2,0,2,4,6, \ldots\},\qquad f(n) =
\begin{cases}
2n & \text{ if } n \text{ is odd} \\ 
-n & \text{ if } n \text{ is even}
\end{cases}$$

I don't know what to do with this though. I never know how to format a proof correctly.