Statement of the problem:
Prove: If a sequence of complex functions $s_n$ on a set $X$ converges uniformly to a complex function $s$, then the sequence of Cesaro means $\sigma_N$ also converges uniformly to $s$.
I have already shown that a sequence of complex numbers $\{s_n\}$ which converges to $s$ has Cesaro means which also converge to $s$, but I'm not sure how to use that to prove this statement. My first instinct was to say that, for each $x$ in the domain, we just have a sequence of complex numbers, so each $\sigma_N(x)$ converges to $s(x)$, but this only establishes pointwise converges (right?). I don't know how to deal with the uniform condition.
Edit: According to my professor, this exercise he assigned is actually false. Can someone come up with an explanation why?
Turns out that the premise is incorrect, as a recent email by my professor has notified.
On $(0,1]$ let $s_1(x) = \frac{1}{x}$ and $s_n = 0$ for $n > 1$, then $s_n \to 0$ uniformly but $\sigma_N \to 0$ pointwise.