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Let $\{f_{n}\}$ be a sequence of nonzero continuous functions on $\mathbb{R}$, which is uniformly bounded, uniformly Lipschitz on $\mathbb{R}$, and the derivative sequence $\{f_{n}'\}$ is also uniformly Lipschitz on $\mathbb{R}$, and $f_{n}\in L^{2}(\mathbb{R})$ for all $n$. If $\{f_{n}\}$ converges uniformly on any closed interval $I\subset \mathbb{R}$ to a continuous function $f$, does this imply that $f\in L^{2}(\mathbb{R})$. If not, what condition(s) the sequence $\{f_{n}\}$ must have to get such result?

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Just curios: are you another incarnation of berry who posted this question: math.stackexchange.com/questions/159507/… ? Or are you just attending the same class? –  user20266 Jun 18 '12 at 16:39

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You need a bound on the $L^2$ norms of the $f_n$. Since you know that your sequence converges pointwise (even better) the same is true for $|f_n|^2$. You can then use Fatou's lemma applied to $|f_n|^2, |f|^2$ to conclude that the limit is in $L^2$.

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if the sequence of $L^2$ norms is unbounded you are probably out of luck. I do admit that I do not have a counterexample right now. If you know that the sequence of $L^2$ -norms is unbounded you seem to have additional information about the $f_n$. Why don't you add this to your question, this may make it easier to provide an answer. –  user20266 Jun 18 '12 at 17:13

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