To show a holomorphic map on the open unit disk is Constant Q. Let $f$ be a holomorphic function on the unit disk $\mathbf{D}$ such that $|f(z)|\rightarrow 1$ as $|z|\rightarrow 1$, and suppose $f(z)\neq 0 \hspace{1ex}\forall z$. Show that $f$ is constant.
Using maximum principle we get $|f(z)|\leq 1$. Now if $1$ is attained we get $f$ is constant. So,we can assume $f:\mathbf{D} \rightarrow \mathbf{D}$. Then  how to proceed? I tried to compose f with an automorphism of $\mathbf{D}$ and applied Schwartz lemma, didn't help much.  

Hint: $|f|$ attains a minimum somewhere.
