$f_n → f$ uniformly on $S$ and each $f_n$ is cont on $S$. Let $(x_n)$ be a sequence of points in $S$ converging to $x \in S$. Then $f_n(x_n) → f(x)$.

Question

Assume that $f_n → f$ uniformly on $S$ and each $f_n$ is continuous on $S$. Let $(x_n)$ be a sequence of points in $S$ converging to $x \in S$. Then $f_n(x_n) → f(x)$.

I'm stuck in thinking about it and I don't manage to prove it neither to find a counterexample. Thanks for your help.

Answer 1

The usual "trick" works well here :

$$\| f_n(x_n) - f(x) \| = \| f_n(x_n) - f(x_n) + f(x_n) - f(x) \|$$

$$ \leq \| f_n(x_n) - f(x_n) \| + \| f(x_n) - f(x) \| $$

$$ \leq \underbrace{\| f_n - f \|_{\infty}}_{A_n} + \underbrace{\| f(x_n) - f(x) \|}_{B_n} $$

And

• $A_n$ converge to $0$ by uniform convergence of $f_n$ to $f$
• $B_n$ converge to $0$ by continuity of $f$

Note that I assumed you already knew that the uniform limit of continuous function is continuous