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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 $\{z_j\}_{j \in \Bbb N} \in \Bbb C$ be a bounded sequence of $\Bbb C$ such that $Re(z_j)>\frac{1}{3}$

Let $$s_m=\sum_{n=1}^\infty \left|\sum_{j=1}^\infty a_{m,j}\left(\frac{1}{(n+1)^{z_j-\frac{1}{m+2}}}-\frac{1}{n^{z_j-\frac{1}{m+2}}}\right)\right|$$

I need to know if $s_m$ is bounded. I just don't know how to solve it. Thanks for any suggestion.

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    because i thougth it as sequences in $\ell^1$, it's the norm of $\ell^1$ – 2017-02-13
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    $\ell^1$ is not a hilbert space. But $\ell^2$ is! – 2017-02-13

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