I have read the proof of Ostrowski's overconvergence theorem (from the Remmert's book 'Classical Topics in Complex Function Theory') and I am confused about one step in the proof.
Let $\sum_{i=0}^{\infty}a_i z^i$ be a power series of a complex variable, that converges compactly on a disc $B$ (so uniformly on any compact subset $K \subset B$).
Let $s_n(z)$ be a sequence of its partial sums. It follows that $s_n(z)$ also converges compactly on $B$. Let $m_i$ be an increasing sequence of natural numbers. Therefore $s_{m_i}(z)$ is a subsequence of $s_n(z)$ for every $z \in B$.
The question is, why does the subsequence $s_{m_i}(z)$ also converge compactly on $B$. I am aware that any subsequence of a convergent sequence is also convergent. I am only confused why the subsequence is also compactly convergent.