# Series of a sub sequence which converges in $\mathbb{C}$

I came across an old question in the analysis course I am studying which goes like this:

Assume that $$(x_n)_{n \in \mathbb{N}}$$ is a sequence which converges to $$0$$ in $$\mathbb{C}$$. Is there always a sub sequence $$(x_{n_k})_{n \in \mathbb{N}}$$ such that the series $$\sum_k x_{n_k}$$ converges absolutely? If yes prove, if not give a counter example.

My first intuition looking at the questions is to that that the answer is yes, but I'm not really sure how to go about proving it. I know that in $$\mathbb{R}$$ if a sequence has a limit and converges, then every sub sequence has the same limit and converges as well. Thus given that the original sequence converges to $$0$$ then each sub sequence must also converge to $$0$$ and the sum of any of these sub sequences would also converge.

Is this correct or am I wrong from the start? Any help is appreaciated

Since $$x_n$$ converges towards zero, for every $$k$$ there exists $$n_k$$ such that $$|x_{n_k}|<{1\over 2^k}$$. $$\sum x_{n_k}$$ converges absolutely.