\text { Let }\left\{f_{n}\right\} \text { be a sequence of continuous real functions that converges uniformly to a function } f \text { on } E \subset \mathbf{R} . \text

{ Prove that } f_{n}\left(x_{n}\right) \rightarrow f(x) \text { for every sequence }\left\{x_{n}\right\} \text { in } E \text { such that } x_{n} \rightarrow x \in E .