Theorem
If {an} is bounded and monotonic then { an} is convergent.
Be cautious to not misuse this theorem. It does not state that if a sequence is not bounded and/or not monotonic that it is divergent. The sequence in that instance was not monotonic but it does converge.
Note: that we can make various variants of this theorem. If {an}is bounded above and increasing after that it converges and similarly if {an} n a is bounded below and decreasing after that it converges.