Problem: Let $(X, ||\cdot ||)$ be some normed linear space. If $\{x_n\} \subseteq X$ is Cauchy, show that $||x_{k+1}-x_k|| < 2^{-k}$ for every $k \in \Bbb{N}$. Attempt: For every $k$, there is $N_k \in \Bbb{N}$ s.t. $||x_n - x_m|| < 2^{-k}$ for every $m,n \ge N_k$. If $k \ge N$, then certainly $||x_{k+1}-x_k|| < 2^{-k}$. But what if $k < N$?