Series Coefficient Convergence implies Uniform Convergence

Question

Trying to find a reference for the following.

Define the entire functions, $$f_n(x)=\sum_{k=0}^\infty a_{n,k}x^k\ \ \ \ \ \ \ \ \ \ \ f(x)=\sum_{k=0}^\infty a_kx^k.$$ If for each $k$, $\displaystyle\lim_{n\to\infty}a_{n,k} =a_k$, then $f_n(x)\to f(x)$ uniformly on compact sets.

Answer 1

Not true. Try $f_n(x) = x^n$ and $f(x) = 0$.

Answer 2

HINT:

It is true if $(f_n)$ is a bounded family, that is, uniformly bounded on compact sets. Use Montel's theorem and the following fact: if from every subsequence of $(f_n)$ you can extract another subsequence that converges to $f$, then $f_n$ converges to $f$.