A problem on sequences and series Let $ (x_n) $ be a sequence of real numbers such that $ \lim x_n =0 $.
Prove that there exists a subsequence $(x_{n_k} )$of $ (x_n) $ such that $ \sum_{k=0}^\infty 2^{k}x_{n_k}$ coverges and $\sum_{k=0}^\infty 2^{k}x_{n_k}$ converges and $\sum_{k=0}^\infty| 2^{k}x_{n_k}|\leq 1$
Didn't have any idea on how to prove or approach it so any help would be appreciated.

Fix a subsequence such that $2^k|x_{n_k}|\leqslant 2^{-k}$. (the sequence $n_k\uparrow \infty$ can be constructed by induction, using the definition of the limit).
