I was going over my practice exam for tonights final, and came across the following problem:

Show that the sequence defined by

a1 = 1

and

an+1 = 3 - 1/an

is increasing and an < 3 for all n. Deduce that {an} is convergent and find its limit.

Any time i try to prove the statement i end up assuming that the sequence is increasing in the actual proof.