Does there exists two nonnegative sequences
$\{a_{n, m} \} $ and $\{r_m\} $ such that
(1). $a_{n, m}\in (\frac{1}{N_o},1)$ for some integer $N_o$ for all $n, m$
(2). $\sum_{n=1}^{\infty}a^2_{n, m}=r_m$
and
$\sum_{m=1}^{\infty}r_{ m} $ converges.
If yes, can we find a family of such sequences?
No. If $\sum_{n=1}^{\infty}a^2_{n, m}$ converges, $a_{n,m}\to 0$ as $n\to \infty$ and thus it cannot be bounded from below.