Ordered field/Convergent sequences/Rules/1/Fact/Proof

Proof

Denote the limits of the sequences by and , respectively. Let be given. Due to the convergence of the first sequence, there exists for

some such that for all the estimate

holds. In the same way there exists due to the convergence of the second sequence for some such that for all the estimate

holds. Set

Then for all the estimate

holds.