Prove that the sequence whose terms are defined recursively by
converges, and compute the limit of the sequence.
Proof. To show the sequence converges we show that it is monotonically increasing and bounded above. To see that it is monotonically increasing we use induction to prove that
For the case we have
Since , the statement holds in the case . Assume then that the statement holds for some positive integer . Then,
since by the induction hypothesis. Hence, so by induction for all positive integers . Hence, the sequence is monotonically increasing.
Next we use induction again to prove the sequence is bounded above by . For we have so the hypothesis holds. Assume then that for all positive integers up to . Then,
Hence, for all positive integers .
This shows that the sequence converges
To compute the limit, assume the sequence converges to a number (we just proved that it converges, so this assumption is valid). Then we have
(We can discard the negative solution since to the quadratic at the end since the sequence is certainly all positive terms.)