Definition: A sequence $\x_n \in \real^d$ is said to be a Cauchy sequence if for every $\eps > 0$ there exists $N$ such that $n,m > N \implies \norm{\x_n - \x_m} < \eps$.
Theorem: In $\real^d$, every convergent sequence is Cauchy, and every Cauchy sequence converges.
Proof: Rudin 3.11.