Let $\{f_{n}\}$ be a sequence of positive continuous functions on $\mathbb R$; $f_{n}:\mathbb R\to \mathbb R$, for all $n\geq 1$, with the folloing properties:
(1) $\{f_{n}\}$ is uniformly bounded by some constant $C>0$,
(2) $\{f_{n}\}$ is uniformly Lipschitz on $\mathbb R$ (so it is uniformly continuous)
(3) $\{f\,'_{n}\}$ is uniformly bounded (sequence of the derivatives), which I think follows from (2).
Does this sequence converges on $\mathbb R$ (or at least contains a subsequence which converges), uniformly on compact sets, or pointwise, to some continuous function $f$ ? If not, what extra (possible) condition must the sequence have to converge to some continuous function?