Infinite real number serie

Question

I am starting on infinite series but I am not quite good yet. I could´nt solve this problem. if you could help that would be awesome.

Prove that if {$a_n$} is a monotone sequence and $\sum_{n=1}^{\infty} a_n$ is a converge serie then $\lim_{n\to \infty} na_n=0$.

Answer 1

The series converges if there exists an $N$ such that for all $m,n$ with $m>n > N$

$|\sum_\limits{n}^{m} a_n| < \epsilon$

let $m = 2n$

Since $a_n$ is monotone

$n|a_{2n}| \le \sum_\limits{n}^{2n} a_n < \epsilon$