Theorem
If {a_{n}} is bounded and monotonic then { a_{n}} 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 {a_{n}}is bounded above and increasing after that it converges and similarly if {a_{n}} n a is bounded below and decreasing after that it converges.