Real numbers/Convergent sequences/Rules/Section


Let and be

convergent sequences. Then the following statements hold.
  1. The sequence is convergent, and

    holds.

  2. The sequence is convergent, and

    holds.

  3. For , we have
  4. Suppose that and for all . Then is also convergent, and

    holds.

  5. Suppose that and that for all . Then is also convergent, and

    holds.

(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.


We give a typical application of this statement.


We consider the sequence given by

and want to know whether it converges and if so, what the limit is. We can not use fact immediately, as neither the numerator nor the denominator converges. However, we can use the following trick. We write

In this form, the numerator and the denominator converges, and the limits are and respectively. Therefore, the sequence converges to .