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For a sequence $\{f_{n}\}$ which converges in $\mathcal{L}^{p}$ space, can we extract a subsequence which is dominated by a function $g \in \mathcal{L}^{p}$?

Can anyone help with this? I thought about using a rapidly Cauchy subsequence but can't get any further...

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    Ok, I think I know how to show that $g$ belongs to $L^p$. Define $g_k=\sum_{j=1}^k|f_{n_{j+1}}-f_{n_j}|$. Then $g_k\to g$ pointwise, $g_k\in L^p$, and $\|g_{k+m}-g_k\|_p\le\sum_{j=k+1}^{k+m}\|f_{n_{j+1}}-f_{n_j}\|_p\le2^{-k-1}$ by the Minkowski's inequality and the choice of $f_{n_k}$. Thus $\int|g_{k+m}-g_k|^p\le2^{(-k-1)p}$, and by Fatou's lemma applied to $k$ fixed and $m\to\infty$, we get $\int|g-g_k|^p\le2^{(-k-1)p}$, and since $\|g\|_p\le\|g-g_k\|_p+\|g_k\|_p$, we get $g\in L^p$.2013-07-11

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If $f_n\to f$ in $L^p$, then there is a subsequence $(f_{n_k})$ such that $\|f_{n_k} - f\|_p < 2^{-k}$. Take $g=|f|+\sum\limits_{k=1}^\infty|f_{n_k}-f|$.

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    @JonasMeyer: Thanks! That observation makes it much easier than what I wrote in the comment to the original question above. Basically, if we define $g_n=\sum_{k=1}^nx_k$, then $\|g_{n+m}-g_n\|=\|\sum_{k=n+1}^{n+m}x_k\|\le\sum_{k=n+1}^{n+m}\|x_k\|$ can be made smaller than arbitrary \epsilon>0 for big enough $n$ and all $m$, and thus $\{g_k\}$ is a Cauchy sequence in a Banach space, hence is convergent to $g=\sum_{k=1}^\infty x_k$.2013-07-11