Let M be a complete metric space. Suppose that $\displaystyle f_n$is a sequence of

continuous functions in $\displaystyle (\mathcal{C}(U, M), \rho)$ which converges to f and $\displaystyle z_n$ is a

sequence in U which converges to a point z in U. Show that$\displaystyle \lim f_n(z_n) = f(z).$

I'm not sure where to start with this one. Clearly, since the functions are continuous, $\displaystyle \lim f(z_n) = f(z)$. I need a way to make sense of the $\displaystyle \rho$ metric so that I can pass the correct limit, but I can't seem to be able to do it.

Can anyone help?

Thank you in advance.