Let f$_n$: E → $R$ be continuous functions for 1 ≤ n ≤ N. Let a$_k$$^n$ be N convergent sequences of numbers and assume $\lim_{k \to inf}$ a$_k$$^n$ = a$_n$. Let f = $\sum_{n=1}^N$a$_n$f$_n$.
I am looking at $\sum_{n=1}^N$ a$_k$$^n$f$_n$. I am pretty sure that it converges pointwise to f but I don't think it converges uniformly to f. However, I am having trouble proving the point wise convergence to f and cannot think of a good counterexample for the uniform convergence to f. Any help would be great!