I am solving some problems in a text, I come across this question. I thought I will not take much of my time on it, but that is not the case.
Question:
Prove that if $\sum \limits_{k=1}^{\infty}\alpha_{k}\phi_{k}$ is convergent whenever $\lim_{k\to \infty}\phi_{k}=0$, then $\sum \limits_{k=1}^{\infty}|\alpha_{k}|\lt \infty$
I am sorry for this question, it seems too simple but I do not know how to tackle it. Please I need just a hint for this.