Prove that: if $a_1 \geq ... \geq … $ and $\sum a_n$ converges, then $lim_{n \rightarrow \infty} n \cdot a_n = 0$
My initial idea was to say that since $\sum a_n$ converges, we know that $lim_{n \rightarrow \infty} a_n = 0$. I'm stuck trying to show that $a_n$ will decrease faster than $n$ increases as $n \rightarrow \infty$. The tools that can be used are sets, the real numbers, sequences of real numbers, and the initial concept of a convergent series. Further notions of series are yet to be defined.
