Series convergence of $\sum_1^\infty{|a_n|}$

Question

Is there anyway to prove that the series $\sum_1^\infty{|a_n|}$ must be convergent if (1) $\lim_{n\to\infty} a_n = 0$ and (2) it is bounded. Could anyone give me an counterexample if not?

An example of that being true is $\sum_1^\infty{\frac{\sin{(4n)}}{4^n}}$

Answer 1

The sequence $\left(\sum_{n=1}^N|a_n|\right)_{N\in\mathbb N}$ is clearly a non-decreasing sequence. If it is also bounded, then it is convergent (every monotonic bounded sequence of real numbers converges).