Math 3333Homework 3 SolutionsName:Peoplesoft ID:Show your work. If a problem requires a proof, explain and justify your steps carefully.Homework papers should be legible and neat, and the pages should be stapled togetherin the correct order. Illegible work may not be graded.Homework should be submittedin classon the indicated due date. Submissions byemail or to the math office will not be accepted.1.TRUE/FALSE:If the statement is true, quote a definition or theorem whichexplains why, or give a proof. If the statement is false, give a counter-example(a) If(sn)is a Cauchy sequence, then(sn)is monotone.False:Letsn=(-1)nn.Then,sn→0, and it follows by Theorem 3(Sec. 4.3) that (sn) is a Cauchy sequence. However, (sn) is not monotone.(b) If(sn)is a convergent sequence and(tn)is bounded, then(sn·tn)isa convergent sequence.False:Letsn= 1 andtn= (-1)n. Sincesn→1, (sn) is a convergentsequence. Also, since|tn| ≤1 for alln∈N, it follows that the sequence(tn) is bounded. However,sntn= (-1)n, which does not converge.

(c) If(sn)and(tn)are sequences such thatsn> tn>1for all positiveintegersn, and (sn·tn) converges, then both (sn) and (tn) converge.False:Let (sn) = (6,4,6,4,6,4, . . .) and lettn= (2,3,2,3,2,3, . . .).Then,sn> tn>1 for alln∈N, and the sequence (sn·tn) = (12,12,12, . . .)converges. However, neither (sn) nor (tn) converge.(d) If(sn)is an unbounded sequence, then every subsequence of(sn)isunbounded.False:Letsn=(nifnis odd,1nifnis even.Then, (sn) =1,12,3,14,5,16, . . .is unbounded.However, the sub-sequencesnk=s2k=12kis bounded since|s2k| ≤12for allk∈N.(e) If (sn) is a bounded, monotone sequence, then (sn) is a Cauchy sequence.True:Assume (sn) is bounded and monotone.By Theorem 1 (Sec.4.3), the sequence (sn) is convergent. Therefore, by Theorem 3 (Sec. 4.3),(sn) is a Cauchy sequence.

(f) If(sn)is a bounded sequence, then(sn)has a Cauchy subsequence.

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