Let $X$ be a topological space and $Y$ a metric space. Let $f_n \colon X \to Y$ be a sequence of continuous functions. Let $x_n$ be a sequence of points of $X$ converging to a point $x \in X$. Suppose that the sequence $(f_n)$ converges uniformly on $X$ to a function $f \colon X \to Y$.
Then how to prove that the sequence $(f_n(x_n))$ converges to $f(x)$ with respect to the metric on $Y$?