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

Can a sequence be Cauchy but not convergent?

A Cauchy sequence need not converge. For example, consider the sequence (1/n) in the metric space ((0,1),|·|). Clearly, the sequence is Cauchy in (0,1) but does not converge to any point of the interval. Definition 8.2.

How do you prove that every Cauchy sequence is convergent?

Let ϵ > 0. Choose N so that if n>N, then xn − a < ϵ/2. Then, by the triangle inequality, xn − xm = xn − a + a − xm < ϵ if m,n>N. Hence, {xn} is a Cauchy sequence.