A sequence in a normed space $X$ is called a Cauchy sequence if and only if for every $\epsilon > 0$ there exists an integer $N\in \Bbb N$, such that $\|x_n-x_m\|\lt \epsilon$ for all $n>m>N$.
Does this mean we can think of an epsilon ball centred on $x_n$ and this contains $x_m$ and if we centred this epsilon ball on any $x_{n+i}$ it would also be guaranteed to contain $x_{m+i}$?
Indeed! $||x_n-x_m|| < \epsilon$ for $n,m > N$ is equivalent to saying that $x_m \in B_\epsilon(x_n)$ for $n,m > N$.