Fix any integer $p\ge2$. Let $\{c_k\}$ be any sequence chosen from the set $\{0,1,2,\dots,p-1\}$. For each $k\in \mathbb N$ define $$x_k:=\sum_{j=1}^k c_j/p^j $$ Prove that $\{x_k\}$ is a Cauchy sequence that converges to some number in $[0,1]$.
I managed to prove that $\{x_k\}$ is Cauchy. I need to prove it converges to some number in $[0,1]$. Please give me some help!