The sequence $$f_n(x):=\sin(x+n), \qquad x\in [0, 2\pi],$$ is relatively compact in $C([0, 2\pi])$ by the theorem of Ascoli-Arzelà. This means that there exist sequences $n_k\in\mathbb N$ and functions $g\in C([0, 2\pi])$ such that $\|f_{n_k}-g\|_\infty\to 0$ as $k\to \infty$.
Can you give some concrete examples of such functions $g$? Is $g=0$ a limit point of $f_n$?
This example could shed some light on the mechanism of proof of the theorem of Ascoli-Arzelà. (Notice that showing that $0$ is a limit point of the numerical sequence $\sin n$ is already nontrivial).
We certainly know that it cannot be the case that $g\equiv0$; the quantity $||f_{n_k}-g||_{\infty}=1$ in that case. I suspect that $g(x)=\sin(x)$ is a concrete example of the functions you are looking for, mostly because we know that $2k\pi$ is equidistributed modulo $1$; there exist $k$ such that $2k\pi$ is arbitrarily close to an integer $n_k$, and so $f_{n_k}$ will be arbitrarily close to $g$.
In fact, using the same kind of argument, you can leverage the fact that the sequence $2k\pi+\alpha$ is also equidistributed modulo $1$ to conclude that $g_\alpha(x)=\sin(x+\alpha)$ is an example for any real $\alpha$.