If $f_n\to f$ pointwise and $f_n(x_n)$ converges, does $\lim f_n(x_n)=\lim f(x_n)$

Question

I have a sequence of real-valued functions $(f_n)_{n\in\mathbb{N}}$ which converge to $f$ pointwise. I also have a sequence of points (which do not necessarily converge) $(x_n)_{n\in\mathbb{N}}$ in a real Hilbert space such that $(f_n(x_n))_{n\in\mathbb{N}}$ converges. Does this imply that $$\lim_{n\to\infty} f_n(x_n)=\lim_{n\to\infty}f(x_n)?$$ I feel like the answer is probably not, but I have not been able to come up with a counterexample yet.

I found this link, but this is a counterexample to a distinct problem (they assume convergence of $x_n$ to a point $x$ and compare $\lim f_n(x_n)$ with $f(x)$; I am asking about the comparison with $\lim_{n\to\infty} f(x_n)$.

Answer 1

Here is a simple example: Let

$$ f_n(x) = \frac{1}{1+(x-n)^2}, \qquad f(x) = 0, \qquad x_n = n. $$

Then $f_n(x) \to f(x)$ for every $x \in \mathbb{R}$, but

$$ \lim_{n\to\infty} f_n(x_n) = 1 \neq 0 = \lim_{n\to\infty} f(x_n). $$