I think you are correct to doubt the validity of saying ... it's not necessarily so.
You might consider applying the Cauchy-Schwartz inequality to . Maybe you can bound it.
Given a covergent series , where each , prove that converges if .
My attempt so far:
I want to say that, if is convergent, then for some N, we have that for all n>N, . The conclusion follows quickly after that, but I do not think that this original statement is necessarily true. For instance, consider
Edit: Just realized that this doesn't explicitly prove my original inequality to be incorrect, but obviously a slight adjustment to this would.