We use math induction When n 1 1 3 1 holds Assume s k s k 1 holds We want to

We use math induction when n 1 1 3 1 holds assume s k

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We use math induction. Whenn= 1, 13-1 holds. Assumesksk+1holds. We wantto showsk+1sk+2, i.e., to show:3-1sk3-1sk+1,which is equivalent to the assumptionsksk+1. We have proved that (sn) is increasing.To show that (sn) is bounded. By the definition,sn+1= 3-1sn3 for alln1. Theboundedness is proved.NowA:= limsnexists. By lettingn→ ∞in both sides ofsn+1= 3-1sn, we obtainA= 3-1A.i.e.,A2-3A+ 1 = 0 which impliesA=3±9-42=3±52.72
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Sinces2= 3-1s1= 3-1 = 2 andsnincreasing, we know thatA2. ThenAcannot be3-52and we must havelimsn=A=3 +52.Theorem 1.21. If(sn)is increasing and unbounded, thensn+.2. If(sn)is decreasing and unbounded, thensn→ -∞.Proof:1. To show: for anyMR, there existsN >0 such thatsnMholds for alln > N.Since{sn}is unbounded and increasing, for theM, there exists an integerN >0 so thatsN> M. Since (sn) is increasing,snsNMholds for alln > N. Therefore,sn+.2. To show: for anyMR, there existsN >0 such thatsnMholds for alln > N.Since{sn}is unbounded and decreasing, for theM, there exists an integerN >0 so thatsN< M. Since (sn) is decreasing,snsN< Mholds for alln > N. Therefore,sn→ -∞.Cauchy sequencesDefinitionA sequence (an) is called aCauchy sequenceif for each>0, there exists apositive integerNsuch that|sm-sn|<,m, n > N.Theorem 1.3Any convergent sequence is a Cauchy sequence.Proof:Supposesns. For any>0, there exists a positive integerNsuch that|sn-s|<2,n > N.For anym, n > N, we then have|sm-sn|=|sm-s+s-sn| ≤ |sm-s|+|sn-s|<2+2=.Thus (sn) is a Cauchy sequence.73
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Theorem 1.4A Cauchy sequence is a bounded sequence.Proof:Since (sn) is a Cauchy sequence, if we take= 1, there exists a positive integerNsuch that|sm-sn|<1,m, n > N.Then|sn|=|sn-sN+1+sN+1| ≤ |sn-sN+1|+|sN+1|<1 +|sN+1|,n > N,which implies|sn| ≤M:= max{1 +|sN+1|,|s1|,|s2|,....,|sN|},n1.This proves (sn) is bounded.Theorem 1.5A sequence is convergent if and only if it is a Cauchy sequence.Proof:If (sn) is convergent, we have proved that (sn) is a Cauchy sequence.Conversely if (sn) is a Cauchy sequence, we want to findssuch thatsns. We’ll provethis in two cases.
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  • Fall '08
  • Staff
  • Mathematical analysis, Natural number, Limit of a sequence, Cauchy sequence

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