Let $a_n > 0$ and for all $n$ let $$\sum\limits_{j=n}^{2n} a_j \le \dfrac 1n $$ Prove or give a counterexample to the statement $$\sum\limits_{j=1}^{\infty} a_j < \infty$$
Not sure where to start, a push in the right direction would be great. Thanks!