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Assume that we have a sequence of non-negative values $x_{n}$ in $\ell_{1}$. Next, assume that the sequence $x_{n}$ converges to $c$ in $\ell_{1}$.

Is the following correct: the sequence is this converge in $\ell_{p}$, for any $p >1$?

Try: the statement is correct, since $||x||_{p} \leq ||x||_{q}$, if $q \leq p$.

LrM
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    Maybe you need to write down a proof of $||x||_{p} \leq ||x||_1$ – reuns Mar 23 '21 at 00:00
  • this statement is from, for example, https://math.stackexchange.com/questions/669360/inclusion-of-lp-space-for-sequences – LrM Mar 23 '21 at 14:11

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Yes, now it is true: it holds that $l_{1}\subsetneqq l_{p}$ for every $p$. Observe that being convergent in $l_{1}$ means the general term $x_{n}$ is infinitesimal as the series $\sum_{n=0}^{+\infty}x_{n}$ is convergent and we apply Cauchy's criterion. This means that for $n$ large enough, $|x_{n}|^{p}<|x_{n}|$, thus if we sum now: $$ \sum_{n=0}^{+\infty}|x_{n}|^{p} < \sum_{n=0}^{+\infty}|x_{n}| $$ Since the series on the right is convergent, so it must do the series on the left, thus it coverges in $l_{p}$ for every $p$.

Davide Trono
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