Show directly from the definition that if (x„) and (yn) are Cauchy sequences, then (X, + yn) and (XµYn) are Cauchy sequences. If X, := n, show that (x„) satisfies lim|xn+1 - Xn| = 0, but that it is not a Cauchy sequence. Let p be a given natural number. Give an example of a sequence (x„) that is not a Cauchy sequence, but that satisfies lim|xn+p - Xn| = 0. %3D
Show directly from the definition that if (x„) and (yn) are Cauchy sequences, then (X, + yn) and (XµYn) are Cauchy sequences. If X, := n, show that (x„) satisfies lim|xn+1 - Xn| = 0, but that it is not a Cauchy sequence. Let p be a given natural number. Give an example of a sequence (x„) that is not a Cauchy sequence, but that satisfies lim|xn+p - Xn| = 0. %3D
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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Transcribed Image Text:Show directly from the definition that if (x„) and (yn) are Cauchy sequences, then (x, + yn) and
(xnyn) are Cauchy sequences.
If xn :=
Vn, show that (x,„,) satisfies lim|x+1 – Xxn| = 0, but that it is not a Cauchy sequence.
Let p be a given natural number. Give an example of a sequence (x„) that is not a Cauchy
sequence, but that satisfies lim|x,+p – xn| = 0.
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