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Exercise 0.9. In class we briefly sketched the construction of the real num- bers using Dedekind cuts. In this problem you will be guided through an alternate construction of the real numbers as equivalence classes of Cauchy sequences of rational numbers. Let Co denote the set of Cauchy sequences of rational numbers. We de- fine a relation - as follows: {an}n€N ~ {bn}neN if {an – bn}neN converges to 0. (1) Prove that the above relation is an equivalence relation. (2) Let {an}neN and {bn}neN be two Cauchy sequences. Prove that {an + bn}neN is a Cauchy sequence. (3) Let {an}neN, {bn}n€N, {Cn}n€EN, and {dn}neN be Cauchy sequences. Assume that {am}n€N ~ {Cn}neN and that {b,}neN that {an + bn}neN ~ {Cn + dn}nɛN- {dm}neN• Prove We then define R to be the set of equivalence classes of Co with respect to the equivalence relation ~. Part (c) shows that the sum of Cauchy sequences descends to a well defined sum between equiv- alence classes; this defines + on R. Similarly, one can do analogous work for multiplication and the order relation (How?)

Exercise 0.9. In class we briefly sketched the construction of the real num- bers using Dedekind cuts. In this problem you will be guided through an alternate construction of the real numbers as equivalence classes of Cauchy sequences of rational numbers. Let Co denote the set of Cauchy sequences of rational numbers. We de- fine a relation - as follows: {an}n€N ~ {bn}neN if {an – bn}neN converges to 0. (1) Prove that the above relation is an equivalence relation. (2) Let {an}neN and {bn}neN be two Cauchy sequences. Prove that {an + bn}neN is a Cauchy sequence. (3) Let {an}neN, {bn}n€N, {Cn}n€EN, and {dn}neN be Cauchy sequences. Assume that {am}n€N ~ {Cn}neN and that {b,}neN that {an + bn}neN ~ {Cn + dn}nɛN- {dm}neN• Prove We then define R to be the set of equivalence classes of Co with respect to the equivalence relation ~. Part (c) shows that the sum of Cauchy sequences descends to a well defined sum between equiv- alence classes; this defines + on R. Similarly, one can do analogous work for multiplication and the order relation (How?)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Exercise 0.9. In class we briefly sketched the construction of the real num- bers using Dedekind cuts. In this problem you will be guided through an alternate construction of the real numbers as equivalence classes of Cauchy sequences of rational numbers. Let Co denote the set of Cauchy sequences of rational numbers. We de- fine a relation - as follows: {an}neN ~ {bn}neN if {an – bn}neN converges to 0. (1) Prove that the above relation is an equivalence relation. (2) Let {an}neN and {bn}neN be two Cauchy sequences. Prove that {an+ bn}neN is a Cauchy sequence. (3) Let {an}neN, {bn}neN, {Cn}n€N, and {dn}neN be Cauchy sequences. Assume that {an}neN ~ {Cn}n€N and that {bn}neN ~ {dn}n€N• Prove that {an + bn}neN ~ {Cn + dn}neN. We then define R to be the set of equivalence classes of Co with respect to the equivalence relation ~. Part (c) shows that the sum of Cauchy sequences descends to a well defined sum between equiv- alence classes; this defines + on R. Similarly, one can do analogous work for multiplication and the order relation (How?)
Transcribed Image Text:Exercise 0.9. In class we briefly sketched the construction of the real num- bers using Dedekind cuts. In this problem you will be guided through an alternate construction of the real numbers as equivalence classes of Cauchy sequences of rational numbers. Let Co denote the set of Cauchy sequences of rational numbers. We de- fine a relation - as follows: {an}neN ~ {bn}neN if {an – bn}neN converges to 0. (1) Prove that the above relation is an equivalence relation. (2) Let {an}neN and {bn}neN be two Cauchy sequences. Prove that {an+ bn}neN is a Cauchy sequence. (3) Let {an}neN, {bn}neN, {Cn}n€N, and {dn}neN be Cauchy sequences. Assume that {an}neN ~ {Cn}n€N and that {bn}neN ~ {dn}n€N• Prove that {an + bn}neN ~ {Cn + dn}neN. We then define R to be the set of equivalence classes of Co with respect to the equivalence relation ~. Part (c) shows that the sum of Cauchy sequences descends to a well defined sum between equiv- alence classes; this defines + on R. Similarly, one can do analogous work for multiplication and the order relation (How?)
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