I am taking complex analysis. There's a question in the book when trying to prove the theorem, and the theorem goes like this:
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 $|z-z_0| \leq R'< R$.
And the question is if we can prove the uniform convergence in every closed subdisk $|z-z_0| \leq R'< R$, then we will have point-wise convergence in the open disk $|z-z_0|<R$. Why is that ? I kind of know that it must be true but cannot give a rigorous proof. Or more precisely, I know that uniform convergence implies point-wise convergence, but how do you deal with the boundary $R$ ?