Let $f_n$ be a sequence of holomorphic functions in a domain $D\subset \mathbb C$.
Let $f_n'$ be the sequence of the derivatives which converges compactly.
And for a $z_0\in D,\ f_n(z_0)$ converges too. Show that $f_n$ converges compactly.
How I started: $f_n(z_0)\to f(z_0)$ for $n\to\infty$, and because $f_n$ is a sequence of holomorphic functions, its limit $f$ will be holomorphic at $z_0$.
That means that $f_n'(z_0)$ converges compactly to $f'(z_0)$ since $f_n'$ converges to $f'$.
If $f_n$ is locally bounded then $f_n$ converges compactly.
How do I show that $f_n$ is locally bounded?