If a sequence is bounded and monotonic then it is convergent, Mathematics

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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.

 


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