The following problem is a Exercise 7,section 1, chapter 6 of Conway's A course in Functional Analysis
Let $A\in {\cal B}(c_0)$ (${\cal B}(c_0)$ is linear bounded operators on $c_0$) and for $n\geq 1$, define $e_n \in c_0$ by $e_n(m)=\delta_{mn}$. Put $\alpha_{mn}=(Ae_n)(m)$ for $n,m\geq 1$. I would like to $(a): M = \sup_m\sum_{n=1}^\infty |\alpha_{mn}|< \infty.$ and $(b): \text{for every}\quad n, \alpha_{mn}\to 0 \quad \text{as}\quad m\to\infty.$
My attempt: I have proved $(a)$
I do not quite understand how to prove $(b)$
Any type of help will be appreciated. Thanks in advance