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My prof. after proving the completeness of $L^p$ space in class using monotone and dominated convergence theorem , he gave us an short proof which he claims to have a blunder mistake and asked us to point it out. It goes like..Let $\sum_k f_k$ be any absolutely convergent series of functions in $L^p$. Then using minkowski inequality $$ || \sum_{k=1}^{n} f_k ||_p \leq \sum_{k=1}^{n} ||f_k||_p $$ Hence $$ || \sum_{k=1}^{n} f_k ||_p \leq \sum_{k=1}^{\infty} ||f_k||_p < \infty$$ and hence $$ || \sum_{k=1}^{\infty} f_k ||_p \leq \sum_{k=1}^{\infty} ||f_k||_p < \infty $$ in limiting case. Thus series $ \sum_k f_k $ converges in p-norm. Hence $L^p$ is complete .

So far I can notice that the limiting case in third equation is wrong because limit of the left hand side may exists or not . Its true for $Lim sup$ but not for limit. Am I pointing out right?

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    $\begingroup$ yes, you are right. $\endgroup$
    – user159517
    Feb 18, 2016 at 23:25

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In fact, $f := \sum_{n=1}^\infty f_n (x)$ does exist (almost everywhere), but one would need to show that.

The other problem with the proof is that we still need to show $\| f - \sum_{\ell =1}^n f_\ell \|_p \to 0$ as $n\to\infty$.

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