Given $f: \mathbb{Z}_5 \to \mathbb{Z}_5$ such that for some fixed $N \in \mathbb{N}$
$$f(x)= x + 5^N (-x_N +x_N^2 +3)$$ I want to show that $f$ is 1-Lipschitz such that $f$ is induced with the p-adic metric.
I started with letting $x \equiv y $ (mod $p^k$). Hence, I need to show that $f(x) \equiv f(y)$ (mod $p^k$), for all $k \in \mathbb{N}$.
I get
\begin{eqnarray*} f(x) -f(y) &=& x-y + 5^N (-x_N + x_N^2 + y_N - y_N^2) \\ &\equiv& 5^N (-x_N + x_N^2 + y_N - y_N^2) \text{ (mod } 5^k)\\ &\equiv& 0 \text{ (mod } 5^k) \end{eqnarray*}
But this is only true for $k \leq N$. How do I show that $f(x) \equiv f(y)$ (mod $p^k$) is also true for $k>N$?
Hint: If $k \gt N$ then $x_N = y_N$ when $x \equiv y \pmod {p^k}$.