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I'm currently having problems with the following exercise, so any help is greatly appreciated.

I,m copying the text literally:

Prove that any monotone sequence bounded in $L^2([0,1])$ strongly converges.

Thank you very much in advance.



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up vote 1 down vote accepted

We can apply monotone convergence theorem. We assume for example that $\{f_k\}$ is decreasing, otherwise consider the sequence $\{-f_k\}$, which will be bounded in $L^2$. Let $f(x):=\lim_{n\to \infty}f_n(x)$, then by Fatou lemma $f$ is in $L^2$. Put $g_n(x):=(f_n(x)-f(x))^2$; each $g_n$ is integrable, $f_n(x)-f(x)\geq f_{n+1}(x)-f(x)\geq 0$ so $\{g_n\}$ is decreasing.

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