A question on point-wise/uniform convergence of a sequence of functions Let $\{f_n(\cdot)\}_{n \in \mathbb{N}}$ be a sequence of real-valued functions defined on $[0, k]$ for some $0 < k < 1$. Suppose that for every fixed $x \in [0,k]$, we have the point-wise convergence
\begin{equation}
\lim_{n \to \infty} f_n(x) = 0.
\end{equation}
The question is: if I let $\{x_n\}_{n \in \mathbb{N}}$ be a sequence of arbitrary points in $[0,k]$. Then does it hold that
\begin{equation}
\sup_{n \in \mathbb{N}} | f_n(x_n) | < \infty?
\end{equation}

No, let $f_{n}:[0,k]\rightarrow\mathbb{R}$ be given by
$$f_{n}(x)=\begin{cases}n&\text{ if }x=\frac{1}{n}\\0&\text{ else}\end{cases}.$$
Clearl pointwise $f_{n}$ converges to $0$, but $$\lim_{n\rightarrow\infty}f_{n}(\frac{1}{n})=\lim_{n\rightarrow\infty}n=\infty.$$
