Let M be a complete metric space. Suppose that is a sequence of

continuous functions in which converges to f and is a

sequence in U which converges to a point z in U. Show that

I'm not sure where to start with this one. Clearly, since the functions are continuous, . I need a way to make sense of the 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.