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# 535 Class Note for MATH 403 at PSU

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This 2 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Pennsylvania State University taught by a professor in Fall. Since its upload, it has received 29 views.

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Date Created: 02/06/15
Math 403 Lecture Notes for Week 3 1 Subsequences De nition 1 Let Ingtn21be a sequence A subsequence of ltfngtn2118 a sequence lt1n1gti21 with ni lt ni1 for all i 2 1 De nition 2 A point y is called a cluster point or accumulation point of a sequence ltfngtn21lf for any 6 gt 0 and any N 2 1 there is an n 2 N such that ly 7 Inl lt e Example 3 The sequence 17 7117 717 H has two cluster points7 1 and 71 Note that the limsup and liminf of a sequence are both cluster points If a sequence has a limit7 then the limit is the unique cluster point of the sequence Proposition 4 The point y is a cluster point of the sequence Ingtn21if and only if there is a subsequence lt1n1gti21 with liminmInl y Proof First7 suppose y is a cluster point of ltfngtn21 We will nd a subsequence whose limit is y First7 letting e g and N 1 we nd an n1 2 1 with lIn1 7 yl lt Next7 letting e i and N n1 17 we nd an n2 gt n1 with lIn2 7yl lt Continuing7 we nd an increasing sequence ni with lInl 7yl lt Since any 6 gt 0 has 6 lt 2711 for some i we see that this subsequence satis es the limit de nition for converging to y Second7 suppose lt1n1gti21 is a subsequence whose limit is y we show y is a cluster point Let 6 gt 0 and N be given We know that there is an M such that for all i 2 M we have lInl 7 yl lt e Choosing 239 large enough so that ni 2 N7 we are done D Hence7 cluster points correspond to convergent subsequences Example 5 Let In n mod 017 ie if n h f where h is an integer and 0 S f lt 17 then In One can show that this sequence is dense in the interval 017 ie for any y E 01 and any 6 gt 0 there is an n with lIn 7 yl lt e Thus any point in 01 is a cluster point of the sequence ltfngtn21 Theorem 6 Bolzano Weierstrass Theorem Every bounded sequence in R has at least one cluster point Proof We will show that a bounded sequence has a convergent subsequence Let Ingtn21be bounded7 and say I gt 0 is such that 7b lt In lt b for all n 2 1 Consider the two intervals L1 7127 0 and R1 012 At least one of these two intervals must contain in nitely many terms of the sequence Choose that interval and let n1 be such that In1 is in that interval Next7 divide that interval in two eeg if the rst interval was L17 we divide it into L2 7127 and R2 7 0 Now one of these two intervals must contain in nitely many terms of the sequence beyond zml Choose this interval and let n2 gt n1 be such that mm is in the chosen intervall We repeat this procedure dividing intervals in two choosing one which contains in nitely many further terms of the sequence and picking one of them to add to our subsequencel At the end we have chosen a subsequence rm with the following property 212 for any i and anyj 2 i we have lznl 7 Injl lt since the two points are in the same subinterval obtained by dividing 7b 12 in halfi times This condition will imply that the subsequence mm is a Cauchy sequence de ned below and the Cauchy Convergence Criterion proved below will show that it therefore converges completing the proof D Spaces which satisfy the Bolzano Weierstrass Theorem are called sequentially compact De nition 7 We say that a sequence lt1nn21is Cauchy if for any 6 gt 0 there is an N such that for any nm 2 N we have lzn 7 zml lt 6 The subsequence de ned above is Cauchy since for a given 6 we can choose N such that 227 lt 6 We start with a preliminary lemma Lemma 8 Nested Intervals Lemma Suppose ltInn21 is a nested sequence of closed bounded nonempty intervals ie each In ambn and In1 Q In for each n Suppose that the lengths of the intervals converges to 0 ie limnnmazn 7 an 0 Then there is a unique real number y with y 6 17211 ie y is in each of the intervals In Proof Since the intervals are nested for any n we have an S an1 S bn1 S bnl Hence the sequence ltan is a monotone increasing sequence bounded above by 121 so it converges say limnn00 an al Similarly the sequence 127 is monotone decreasing and bounded below by a1 so we have limnn00 bn 12 We claim that a 12 This follows since an S a S b S In for all n so I 7 a S In 7 an since limnnmazn 7 an 0 we must have b 7 a 0 Let y a 12 We see that y is in each of the intervals In since an S a y b S bnl We nally check that this y is unique Suppose we have another point 2 6 17211 say y lt 2 Again we have an S y lt 2 S In for each n so 27yltbn7anandweseethatz7y0sozyl D Theorem 9 Cauchy Convergence Criterion A sequence ltznn21converqes in R if and only it is Cauchy Proof We want to produce a nested sequence of intervals to which to apply the lemma Let lt1nn21be Cauchy and let 6k Applying the de nition we

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