induction only finishes ?? And x-y divides P(x) - P(y)

Quidditch wrote: Prove that for all polynomials $P \in \mathbb{R}[x]$ and positive integers $n$, $P(x)-x$ divides $P^n(x)-x$ as polynomials. Note that \[P(x)-x\mid \left(P^n(x)-P^{n-1}(x)\right)+\hdots+(P(x)-x)=P^n(x)-x\]