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**PRLM** Let $\displaystyle f_n$ be integrable on [0,1] for all $\displaystyle n$ and $\displaystyle f_n\rightarrow f$ uniformly. Show that $\displaystyle f$ is integrable and $\displaystyle \int f_n dx \rightarrow \int fdx$.

Since $\displaystyle f_n$ converges uniformly to $\displaystyle f$, for any $\displaystyle \epsilon >0$, there is $\displaystyle N$ such that for all $\displaystyle x$ and for all $\displaystyle n \geq N$, $\displaystyle |f-f_n|<\epsilon$. And since $\displaystyle |f|-|f_n| \leq|f-f_n|$, $\displaystyle \int |f|dx-\int |f_n|dx \leq \int |f|-|f_n|dx \leq \int |f-f_n|dx \rightarrow 0$.

So $\displaystyle \int |f| = lim \int |f_n|$.

i have a question here. i know that for each $\displaystyle n$, $\displaystyle \int |f_n| dx < \infty$ but how can i show that $\displaystyle lim \int |f_n|dx < \infty$.

help would be appreciated so much.