Let $f:D \rightarrow D$ be a holomorphic function such that f has a zero of order $N\geq 1$ at the origin, then show that $|f(z)|\leq |z|^N$

Question

Let $f:D \rightarrow D$ be a holomorphic function such that f has a zero of order $N\geq 1$ at the origin, then show that $|f(z)|\leq |z|^N$ $\forall z\in D$, where $D=\{z\in\mathbb{C}| |z|<1\}$

I tried to use Schwarz's lemma inductively here. I showed that $f^n(D)\subseteq D$, but then I don't know how to approach it