Transcription of Theorem (The Monotone Convergence Theorem)
1 Math 410 Section : The Monotone Convergence Theorem1. Idea: We know that if a sequence converges then it must be bounded. We also know the reverse is nottrue. However in the case of Monotone sequences it Definitions: We say{an}is monotonically ( Monotone ) increasing if n, an+1 an. We say{an}is monotonically ( Monotone ) decreasing if n, an+1 an. A sequence is Monotone if it is (The Monotone Convergence Theorem ):If{an}is Monotone and bounded then it converges. In addition if this is the case then: If is Monotone increasing then it converges to sup{an|n N} If it is Monotone decreasing then it converges to inf{an|n N}Intunition:For example if a sequence is Monotone increasing and has an upper bound then eventually it mustlevel off. Notice that this doesn t have to happen at the upperbound, it could happen :Suppose{an}is Monotone increasing.
2 DefineSto be the set of terms in{an}and defineL= sup(S)which exists by the Completeness Axiom sinceSis bounded. We claim{an} >0, SinceL is not an upper bound forS(sinceLis the least upper bound) we know there issomeNsuch thataN> L and moreover since{an}is increasing for alln Nwe havean> L .However sinceLis an uppoer bound we havean L < L+ as well and hence for alln Nwe haveL < an< L+ and so|an L|< .The proof for monotonically decreasing is (a) Warning: We can t conclude the sequence converges to the bound. For example{1n}is monotonedecreasing and bounded below by 17 but it certainly doesn t converge to 17.(b) Example: Consider the sequence defined byan=n k=312kk2 This sequence is Monotone increasing and for allnwe havean=n k=312kk2<n k=312k< k=312k=14(Geometric Series)Thus it converges.
3 As with the warning above we cannot conclude the sequence converges to14.(c)Corollary:Letc Rwith|c|<1. Then{cn} :Suppose 0< c <1. The sequence is monotonically decreasing and bounded below by 0. Ittherefore converges toL= inf{cn|n N}so we claimL= 0. SupposeL >0 then observe thatfor allnwe havecn=cn+1c Lc> Lwhich contradictsLbeing the proof for 1< c <0 is similar, the proof forc= 0 is
