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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

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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.

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