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}

The question has been answered. To answer a question the OP asks in comments on the answers: It's easy to  see that $\sup_nf_n(x_n)<\infty$ for every sequence $x_n$ if and only if the $f_n$ are uniformly bounded ($f_n(x)\le C$ for every $n$ and every $x$.)
And, since it seems possible that this is what you really want to know: It's also easy to see that $f_n(x_n)\to0$ for every sequence $(x_n)$ if and only if $f_n\to0$ uniformly.
