If $\{f_{n}(x)\}$ is a sequence of continuous functions on $\mathbb R$, with $|f_{n}(x)|\leq C, \forall n$, and $\lim_{n\to\infty}f_{n}(x)=0$ uniformly on $\mathbb R$, does there exist a subsequence of $\{f_{n}\}$ which is decreasing on $\mathbb R$?
Edit: What if $\{\sup_{\mathbb R}|f_{n}(x)|\}$ is also converges to 0, and all $f_{n}$ are positive continuous functions with the above properties?