Sequence of square integrable functions 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?

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 $L^2$.
