Let f be analytic on an open disc . If , then f is bounded on the circle | . My notes make use of this fact but it doesn't prove it so I have tried to do it myself, is this correct?

f is analytic on , so has a power series expansion with radius of convergence (by Taylor).

So on the circle:

, which converges since r = |r| < R'. Hence f is bounded on the circle - is this ok? Thanks for any help and advice