Let $K$ be a compact of $\mathbb C$ and $S$ be a subset of $K$. Suppose that one has $p\in\mathbb N$ sequences of functions $(f_{n,i})_n$ ($1 \le i\le p$) uniformly convergeing towards a function $f_i$ on $S$. Suppose that one has $p$ functions $g_1,\cdots,g_p$ continuous on $K$. Can one assert that the sequence of functions $\left(\sum_{i=1}^pg_if_{n,i}\right)_n$ converges uniformly towards the function $\sum_{i=1}^pg_if_i$ on $S$?
Thanks in advance for any answer