Let $\{a_{n,m}\}_{n,m \in \Bbb N} \in \Bbb R_+$ a double real positive sequence such that $$b_m=\sum_{n=1}^\infty a_{n,m}<\infty$$ and such that $$\lim_{m \to \infty} a_{n,m}=0$$
My question is if the following relation is true:
$$\sup_{m \in \Bbb N}(b_m)<\infty$$
Thanks for any suggestion.
No.
Write $1[\phi]$ for the indicator function, whose value is $1$ if $\phi$ is true, and $0$ otherwise.
Let $$a_{n,1} = 1[n=0]$$ $$a_{n,2} = 1[1 \leq n \leq 2]$$ $$a_{n,3} = 1[3 \leq n \leq 6]$$ $$a_{n,4} = 1[7 \leq n \leq 10]$$ and so on, taking $a_{n,m}$ to be $1$ if $n$ is between the $m-1$th and $m$th triangular number, and $0$ otherwise.
Then $b_m = m$, which is finite but tends to $\infty$ as $m \to \infty$; but $$\lim_{m \to \infty} a_{n,m} = 0$$ because for fixed $n$, $a_{n,m}$ is eventually $0$.