I have that $\ell^{1}$ with the norm defined by $\left\|\left(a_{1}, a_{2}, \ldots\right)\right\|_{1}=\sum_{k=1}^{\infty}\left|a_{k}\right|$. I am trying to prove that this is a Banach space. That is I'm trying to prove that it is a complete normed vector space. So we assume that we have a Cauchy sequence $(x_k)$. If I have $\left\|x_{n_{k}}-x_{n_{k+1}}\right\| \leq 2^{-k}$ then am i done because $\sum_{k}\left\|x_{n_{k}}-x_{n_{k+1}}\right\| \leq \sum_{k} 2^{-k}<\infty$ converges. That is $(_{})$ converges. Now, a Cauchy sequence that has a converging subsequence is convergent.
If this is right, how can i prove that $$\left\|x_{n_{k}}-x_{n_{k+1}}\right\| \leq 2^{-k}$$