theorem if an is bounded and monotonic then an

Question

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.