Every Cauchy sequence of real numbers is bounded, hence by Bolzano–Weierstrass has a convergent subsequence, hence is itself convergent.

How do you prove that every Cauchy sequence is bounded?

Lemma: Every Cauchy sequence is bounded. Proof: Let (an) be Cauchy. We choose 0<ϵ0. So ∀n>m≥N0 we have that |an−am|<ϵ0.

Are all Cauchy sequences convergent?

Theorem. Every real Cauchy sequence is convergent. Theorem. Every complex Cauchy sequence is convergent.