Theorem: "If f is analytic in the disk |z-z_0| < R, then the Taylor Series converges to f(z) for all z in the disk. Furthermore, the convergence of the series is uniform in any closed subdisk |z-z_0| <= R' < R
Proof: It is enough to prove uniform convergence on all closed subdisks |z-z_0| <= R' (why?)
Can someone help in answering the (why?) part of the proof. Many thanks in advance!
Why? Because for each $z$ in the disc $|z-z_0|<R$ there exists a positive number $R' < R$ such that $z$ is contained in the closed subdisc $|z-z_0| \le R'$: simply take $$R' = \frac{|z-z_0| + R}{2} $$