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.