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?
Sequence of square integrable functions
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real-analysis
sequences-and-series
convergence
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0Just curios: are you another incarnation of berry who posted this question: http://math.stackexchange.com/questions/159507/sequence-of-lipschitz-functions/159520#159520 ? Or are you just attending the same class? – 2012-06-18