Given $\epsilon > 0$ and sequence $\{a_k\}_{k=1}^\infty$ where $a_k > 0$ for all $k$ and $$\sum_{k=1}^\infty a_k = A > \epsilon,$$ I know that I can find a finite set $K\subset \mathbb{N}\ \ |\ \ |K| = N,$ such that $$\sum_{k\notin K}a_k\leq \epsilon. $$ My question is if I can put a loose bound on $N$? My intuition is that for most $k\in \mathbb{N},$ $a_k \leq \frac{1}{k},$ and so I've hoped I could somehow show that I can find $K$ such that $N = O\left(\frac{1}{\epsilon}\right).$
My apologizes if its a trivial question, I don't remember much of calculus/analysis.