MATH 314 001 – Test #1 – 9/29/06 1Fri Sep 29 10:04:57 MDT 2006 /m314.fa06/handouts314/t1 314 929/t1 314 929This test has pages 1 – 2. Take a moment to make sure you have them all.1 Ten points each:(a) State the formal definition of lim(sn) = L for sequence (sn)∞n=1. The best answerhere uses logical symbols such as ∃, ∀, and =⇒.(b) Write down the formal statement of the Archimedean Property of Order in R. Againthe best answer uses logical symbols such as ∃, ∀, and =⇒.(c) Write down the definition of the sentencez is an upper bound of set A.The best answer also uses logical symbols.(d) Write down the formal negation, or denial, of your part-(c) answer. This means thatyou will be writing down what is true if it is not the case that z is an upper bound ofA.2 Write up a formal mathematical-induction proof of the assertionnXk=1(2k − 1) = n2.Write so as not to attract the attention of the Circular-Reasoning Police. Your proof mustbe a formal mathemtical-induction proof. (20 points)3 Let the sequence (an)∞n=1be given byan=30n + 56n − 29.Write a proof of the value of lim(an). Your proof must work directly from the definitionof lim(an). (20 points)MATH 314 001 – Test #1 – 9/29/06 24 Pick one of the following assertions (a)-(d)(say which) and write up a formal proof of thechosen assertion.(a) Let (an)∞n=1and (bn)∞n=1be convergent sequences. Prove that (cn)∞n=1is also aconvergent sequence, where cn= an+ bn. (20 points)(b) Let S and T be nonempty bounded subsets of R. Suppose, moreover, that s 6 twhenever s ∈ S and t ∈ T . Prove thatsup(S) 6 inf(T ).(20 points)(c) Let (an)∞n=1be a sequence and let lim(an) = L, where L ∈ R.Suppose also that M ∈ R and that, for all n, an< M.Prove that L 6 M. (25 points)(d) Let (an)∞n=1, (bn)∞n=1, and (zn)∞n=1be sequences.Suppose that (an) and (bn) are convergent sequences with identical limits.Suppose also that there exists a natural number K such that n > K impliesan6 zn6 bn.Prove that (zn) is also a convergent sequence. (25 points)Fri Sep 29 10:04:57 MDT 2006 /m314.fa06/handouts314/t1314 929/t1 314