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> Consider a sequence of functions $f\_n$ defined on $[0,1]$ such that $f\_n=\frac{x^n}{1+x^n}$. Is it true that $\lim\_{n\to \infty} \int\_{0}^1 f\_n(x)dx=\int\_0^1 \lim\_{n\to \infty} f\_n(x)dx$?
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My first attempt is to prove that $\{f\_n\}$ uniformly converges to some $f$. But it is not. Is there any way to do this?