Convergence in infinite double sequence

Question

I was told in math class that with $\sum\limits_{n=1}\limits^{\infty}\lvert{a_{mn}}\rvert = b_m$ and $\sum\limits_{m=1}\limits^{\infty}b_m<\infty$, we can conclude that $\sum\limits_{m=1}\limits^{\infty}a_{mn}<\infty$. I don't really see the connections, and am struggling to find out why this is the case. Why does this work?

Answer 1

The comment above by zkutch helped a lot. Since $\lvert a_{mn}\rvert \leq b_m$ and $\sum\limits_{m=1}\limits^{\infty}b_m<\infty$, $\sum\limits_{m=1}\limits^{\infty}\lvert a_{mn} \rvert < \infty$ and thus $\sum\limits_{m=1}\limits^{\infty}a_{mn}$ converges.