# pointwise converging sequence of continuous functions on a compact: bounding oscilation from below

Assume It is known know that oscilation $\omega(f_n, U) > \epsilon$ for continuous functions $f_n$ and some open set $U$ starting from some $N$. Also it is known that $f_n \to \varphi$ pointwise. It is also safe to assume that $\overline{U}$ is compact and that $f_n$ is uniformly bounded.

Here $$\omega(f,U) = \sup_{x,y \in U} \Big|f(x) - f(y)\Big|$$

Intuitively, it seems true that that $\omega(\varphi,\overline{U}) \ge \epsilon$.

But is it actually true?

I thought that it must be possibele to find points $x_n,y_n \in \overline{U}$ such, that
$$\omega(f_n,\overline{U}) = \Big|f_n(x_n) - f_n(y_n) \Big|$$

Then compute limits of converging subsequences, which exist by compactness of $\overline{U}$. Name them $Y$ and $X$.

And Then to show that $|\varphi(Y) - \varphi(X)|$ is substantially big by convergence and continuity arguments. But I don't know how to do it as both argument and function depend on $n$.