1. The problem statement, all variables and given/known data Let (xn)n[itex]\in[/itex]ℕ and (yn)n[itex]\in[/itex]ℕ be Cauchy sequences of real numbers. Show, without using the Cauchy Criterion, that if zn=xn+yn, then (zn)n[itex]\in[/itex]ℕ is a Cauchy sequence of real numbers. 2. Relevant equations 3. The attempt at a solution Here's my attempt at a proof: Let (xn) and (yn) be Cauchy sequences. Let (zn) be a sequence and let zn=xn+yn. Since (xn) and (yn) are Cauchy, [itex]\exists[/itex]N[itex]\in[/itex]ℕ such that, |xn-xm|<ε/2, and |yn-ym|<ε/2 for n,m≥N. Let n,m≥N and let zn,zm[itex]\in[/itex](zn). Then, |zn-zm|=|xn-xm|+|yn-ym| <ε/2+ε/2=ε. Therefore, |zn-zm|<ε for all n,m≥N and hence, (zn) is a Cauchy sequence of real numbers. Is this correct? Any input is appreciated. Thanks.