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

where

Edit: Just realized that this doesn't explicitly prove my original inequality to be incorrect, but obviously a slight adjustment to this would.