How to Prove Uniform Convergence?

Question

I was trying to prove If f is holomorphic on the open unit disk and|f(z)|= 1 for|z|= 1, then f is a finite Blaschke product. In the proof, I saw Since|f(z)|→1 uniformly as|z|→1, there is an r <1 so that f is nonvanishing on the annulus r≤|z|<1. My question is why does |f(z)| tends to 1 uniformly? And why does f has a finite number of zeroes in the open unit disk?`