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(*sup) : Every nonempty set A of real numbers which is bounded from above has a supremum. Prove that this property implies that every Cauchy sequence {xn} of real numbers has a limit. Please follow the steps listed below to finish the proof. (*inf) : Every nonempty set A of real numbers which is bounded from below has a infimum. Write YN sup{*n : n > N}, zN = inf{xn :n < N}. Using (*sup) and (*inf) to prove that {yN} and {zN} both have limits, and hence prove that the Cauchy sequence {xn} has a limit.

(*sup) : Every nonempty set A of real numbers which is bounded from above has a supremum. Prove that this property implies that every Cauchy sequence {xn} of real numbers has a limit. Please follow the steps listed below to finish the proof. (*inf) : Every nonempty set A of real numbers which is bounded from below has a infimum. Write YN sup{*n : n > N}, zN = inf{xn :n < N}. Using (*sup) and (*inf) to prove that {yN} and {zN} both have limits, and hence prove that the Cauchy sequence {xn} has a limit.

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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In this problem, we only assume the following property for R: (*sup) : Every nonempty set A of real numbers which is bounded from above has a supremum. Prove that this property implies that every Cauchy sequence {xn} of real numbers has a limit. Please follow the steps listed below to finish the proof. (*inf) : Every nonempty set A of real numbers which is bounded from below has a infimum. Write YN = sup{xn : n > N}, zN = inf{xn :n < N}. Using (*sup) and (*inf) to prove that {yN} and {zN} both have limits, and hence prove that the Cauchy sequence {xn} has a limit.
Transcribed Image Text:In this problem, we only assume the following property for R: (*sup) : Every nonempty set A of real numbers which is bounded from above has a supremum. Prove that this property implies that every Cauchy sequence {xn} of real numbers has a limit. Please follow the steps listed below to finish the proof. (*inf) : Every nonempty set A of real numbers which is bounded from below has a infimum. Write YN = sup{xn : n > N}, zN = inf{xn :n < N}. Using (*sup) and (*inf) to prove that {yN} and {zN} both have limits, and hence prove that the Cauchy sequence {xn} has a limit.
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