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Suppose that $f_n$ is a sequence of functions in $C[0,1]$ and that $f_n$ converges uniformly to $f$ on $[0,1]$. Is it true that $\int_0^{1-1/n} f_n$ $\rightarrow$ $\int_0^1f$?

I think the answer is yes. Since $f_n$ converges uniformly to $f$, so $\int_0^{1} f_n$ $\rightarrow$ $\int_0^1f$. But $\int_0^{1-1/n} f_n$ = $\int_0^{1} f_n + \int_1^{1-1/n} f_n$. So we just need to prove that $\int_1^{1-1/n} f_n$ converge to $0$. But I don't know how to do it. Or if I'm even on the right track.

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Since the $f_n$ are uniformly convergent, they are also uniformly bounded. That observation should do the trick.

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