# N 1 sup s of course the same applies to the infimum

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n!1supS.Of course, the same applies to the infimum, except that the sequence will be monotonedecreasing.Another interesting application (done in class) is constructing sequences which convergemonotonically topa, for anya >0. See Example 3.3.5 in the text.Exercise 2.13.We return to Example 2.1 (c),x1= 2andxn+1=12xn+2xn,n= 1,2,3,. . . .to showlimn!1xn=p2.(a) Use induction to show thatx2n-20for alln2N.(b) Use induction to show that(xn)is monotone decreasing inn.(c) Use the Monotone Sequence Theorem to show convergence,xn---!n!1x, and identifyxas the solution of a polynomial equation.About Induction.We hope that you’ve already seen mathematical induction somewhereelse before. But we remind you of it here, and in the spirit of healthy skepticism about allthings, (which we encourage in studying math,) we show that it isn’t magic.Proposition 2.14.For eachn2N, letP(n)denote some statement involving the value ofn. If we can show both:21
(1)P(1)is true; and(2) For everyn2N, ifP(n)is assumed to be true thenP(n+ 1)is true,thenP(n)is true for alln2N.Proof.LetS={n2N|P(n) is false.}. To derive a contradiction, we suppose thatSis notempty. By the Well Ordering Principle (Corollary 1.13),Shas a minimal element. Call theminimal element (k+ 1), so (k+ 1)2SNbut for anynk,n62S. Therefore,P(k) istrue. But, by (2),P(k) true impliesP(k+1) is also true, and this contradicts (k+1)2S.From the proof, we can see that the following version of induction is also verified:If we can show both:(1)P(1)is true; and(2) For everyk2N, ifP(n)is assumed to be true for allnk, thenP(k+ 1)is true,thenP(n)is true for alln2N.Sometimes this form of induction is needed (for example, for sequences defined by iterationinvolving several previous terms.) This is discussed in section 1.2 of the textbook.22
2.3Divergent Sequences and SubsequencesAlthough we prefer sequences which converge, there are also many interesting things to learnabout divergent sequences.Sequences can diverge in various ways, some more interestingthan others.The simplest kind of divergence is calledproper divergencein the book.Definition 2.15(Properly divergent sequences).Let(xn)n2Nbe a sequence inR.(a) We say the sequence properly diverges to+1if:8a >09K2Nso thatxn> a8nK.We writelimn!1xn= +1orxn---!n!1+1, even though the sequence does not converge.(b) We say the sequence properly diverges to-1if:8b <09K2Nso thatxn< b8nK.We writelimn!1xn=-1orxn---!n!1-1, even though the sequence does not converge.So a sequence diverges to infinity if, for any fixed numbera >0,xniseventually alwayslarger thana.Example 2.16.Letxn=pn(3 + sin(n)),n2N. Thenxn---!n!1+1.Choose anya >0. Since sin(n)≥ -1 for anyn, we havexnpn(3-1) = 2pn > aprovidedn >a24. So, takeK2NwithK >a24. IfnK >a24, by the above calculation,xn> a, and by definitionxn---!