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Let $\{a_{n,m}\}_{n,m \in \Bbb N}$ be a double complex sequence such that:

$$\lim_{m \to \infty} \sum_{n=1}^\infty |a_{n,m}|^2 =0$$ and $$\lim_{m \to \infty} \sum_{n=1}^\infty a_{n,m}$$ is convergent.

I would like to know if the following relation is true

$$\lim_{m \to \infty} \sum_{n=1}^\infty a_{n,m}=0$$

Thanks

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    double sequence i mean for each $m$, $a_n,m$ is a complex sequence – 2017-02-03
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    @JackD'Aurizio the limit is zero not the sum – 2017-02-03

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It is not. If we choose $a_{n,m}=\frac{1}{m}$ for every $1\leq n\leq m$ and $0$ otherwise, we have: $$ \sum_{n\geq 1} |a_{n,m}|^2 = \frac{1}{m},\qquad \sum_{n\geq 1} a_{n,m}=1.$$

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    Simple and elegant. – 2017-02-03
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    thanks @Jack D'Aurizio for this counterexample – 2017-02-03