Question

State and prove Cauchy theorem on limits

Answer 1

Definition We say that a sequence of real numbers {an} is a Cauchy sequence provided that for every ϵ > 0, there is a natural number N so that when n, m ≥ N, we have that |an − am| ≤ ϵ. ... Theorem 1 Every Cauchy sequence of real numbers converges to a limit. Proof of Theorem 1 Let {an} be a Cauchy sequence.