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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?

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By Arzela-Ascoli's theorem your sequence has a uniformly convergent subsequence (the limit of which will, of course, be continuous) on each compact subset. You cannot hope for convergence of the sequence itself, though.

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    @Berry yes, that is correct.2012-06-17