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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.