Uniformity on a closed unit disk

Question

Let $(f_n)^\infty_{n=1}$ be a sequence of functions, which are all analytical on the open unit disk $\{z \in \mathbb{C} : |z| < 1\}$ and continuous on the closed unit disk $\{z \in \mathbb{C} : |z| \leq 1\}$. Suppose that $f$ has the same characteristics and that $f_n \to f$ is uniform on the unit circle $\{z \in \mathbb{C} : |z| = 1\}$. Proof that $f_n\to f$ is uniform on the closed unit disk.

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What I've tried so far:

I know that uniformity demands that $\forall \epsilon > 0\ \exists N \in\mathbb{N}_{>0} : \forall n > N: |f_n(x)-f(x)|< \epsilon$. Since the $f_n$

Answer 1

Hint: Maximum modulus principle.