[Math] Proving that if the partial sums are bounded (i.e. $|\sum_n^N a_n|$ is bounded) and that $\sum |a_n|^2$ converges, then $\sum a_n$ converges.

Question

I am wondering if the following is true and if so need help proving it. If the series of partial sums is bounded, that is $|\sum_{n=1}^N a_n|$ is a bounded sequence indexed by $N$ and that $\sum_{n=1}^{\infty} |a_n|^2$ converges, then $\sum_{n=1}^{\infty} a_n$ converges. The $a_n$ are complex.

Thanks!

Answer 1

Look at $$1-\frac{1}{2}-\frac{1}{2}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}-\frac{1}{8}-\frac{1}{8}-\frac{1}{8}-\frac{1}{8}-\frac{1}{8}-\frac{1}{8}-\frac{1}{8}-\frac{1}{8}+\frac{1}{16}+\cdots.$$ The series does not converge. But $1$ is an upper bound for the partial sums, and $0$ is a lower bound, and $\sum a_n^2=2$.