Real numbers/Convergent sequences/Rules/Fact/Proof

Proof

(1). 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.


(2). Let be given. The convergent sequence is bounded, due to fact, and therefore there exists a such that for all . Set and . We put . Because of the convergence, there are natural numbers and such that

These estimates hold also for all . For these numbers, the estimates

hold.

For the other parts, see exercise, exercise and exercise.