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Let $(f_{n})_{n}$ a sequence in $\mathcal{L}^1(\mathbb{R})$ and $f_{n}. Also $\int_{\mathbb{R}}f_{k}^-dm < \infty$ for some $k\in \mathbb{N}$. Show that $$\lim_{n\to\infty}\int_{\mathbb{R}}f_{n}dm=\int_{\mathbb{R}}\lim_{n\to\infty}f_{n}dm.$$

I think write this like a growing sequence for use the monotone Lebesgue theorem, some help?

Thanks!

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