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Let $(f_{n})_{n\in\mathbb{N}}$ be a sequence of integrable functions and $f$ an integrable function. I have to show that, if $$ \sum_{n\in\mathbb{N}}\int{|f_{n}-f|d\mu}<\infty, $$ then $f_{n}\rightarrow f$ almost everywhere. The inequality written before implies that $\lim_{n}\int{|f_{n}-f|}=0$, but I do not know how to continue from that.

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Since $(|f_n-f|)_{n\geqslant 1}$ is a sequence of non-negative functions we have $$ \int \sum_{n\geqslant 1} |f_n-f| \,\mathrm d\mu=\sum_{n\geqslant 1}\int |f_n-f|\,\mathrm d\mu<\infty. $$ Therefore, $\sum_{n\geq 1}|f_n-f|<\infty$ almost surely and hence also $\lim_{n\to\infty}|f_n-f|\to 0$ almost surely.

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