We use math induction. Whenn= 1, 1≤3-1 holds. Assumesk≤sk+1holds. We wantto showsk+1≤sk+2, i.e., to show:3-1sk≤3-1sk+1,which is equivalent to the assumptionsk≤sk+1. We have proved that (sn) is increasing.To show that (sn) is bounded. By the definition,sn+1= 3-1sn≤3 for alln≥1. 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

Sinces2= 3-1s1= 3-1 = 2 andsnincreasing, we know thatA≥2. 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 anyM∈R, there existsN >0 such thatsn≥Mholds for alln > N.Since{sn}is unbounded and increasing, for theM, there exists an integerN >0 so thatsN> M. Since (sn) is increasing,sn≥sN≥Mholds for alln > N. Therefore,sn→+∞.2. To show: for anyM∈R, there existsN >0 such thatsn≤Mholds for alln > N.Since{sn}is unbounded and decreasing, for theM, there exists an integerN >0 so thatsN< M. Since (sn) is decreasing,sn≤sN< 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:Supposesn→s. 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

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|},∀n≥1.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 thatsn→s. We’ll provethis in two cases.

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- Mathematical analysis, Natural number, Limit of a sequence, Cauchy sequence