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Let $\{a_{m,j}\}_{m,j \in \Bbb N} \in \Bbb C$ such that for each $m$ the sequence $a_{m,j} \in c_{00}$ definitely zero sequence and for each $j$ we have $\lim_{m \to \infty}a_{m,j}=0$

Let $\{j_n\}_{n \in \Bbb N} \in \Bbb N$ a sequence of $\Bbb N$

I would like to know if $$\sup(|a_{m,j_m}|)_{m \in\Bbb N}<\infty$$ Thanks for any suggestion.

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    What do you mean by "definitely zero sequence"? Would a counterexample be $a_{n,n} = n$, $a_{n,j} = 0$ otherwise? – 2017-02-13
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    The answer is no. You would need to have some kind of "uniform convergence" among the rows and/or columns – 2017-02-13
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    @RobertIsrael exist $k$ such that $a_{m,j}=0$ for each $j>k$ – 2017-02-13
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    @RobertIsrael a sequence with at most finitely many non-zero elements. So yes it would. – 2017-02-13
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    thanks @Omnomnomnom , can you tell me what kind of uniform convergence – 2017-02-13
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    @MateyMath we'd need to be able to say something like: "for all $\epsilon > 0$: there exists an $m_0$ such that **for all j**: $m < m_0 \implies |a_{m,j}| < \epsilon$" – 2017-02-13
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    ok @Omnomnomnom thanks for this – 2017-02-13

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