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2. Let (2n) and (yn) be Cauchy sequences of real numbers. Define (Tn) to be equivalent to (yn), written (n)~ (yn), if lim Tn - yn = 0. 14x Show that this defines an equivalence relation on the set of all Cauchy sequences of real numbers. Furthermore, prove that if (rn) and (yn) are Cauchy sequences, then ||æn - Yn|-|Tm - Ym|| ≤ n - Tm+ Ym-Yn- Conclude that (n - Yn) is a Cauchy sequence.

2. Let (2n) and (yn) be Cauchy sequences of real numbers. Define (Tn) to be equivalent to (yn), written (n)~ (yn), if lim Tn - yn = 0. 14x Show that this defines an equivalence relation on the set of all Cauchy sequences of real numbers. Furthermore, prove that if (rn) and (yn) are Cauchy sequences, then ||æn - Yn|-|Tm - Ym|| ≤ n - Tm+ Ym-Yn- Conclude that (n - Yn) is a Cauchy sequence.

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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2. Let (an) and (yn) be Cauchy sequences of real numbers. Define (rn) to be equivalent to (yn), written (xn) ~ (Yn), if lim n – Yn| = 0. Show that this defines an equivalence relation on the set of all Cauchy sequences of real numbers. Furthermore, prove that if (an) and (Yn) are Cauchy sequences, then ||an – Yn| – |æm – Ym|| < |æn – m| + |Ym – Yn|- Conclude that (|æn – Yn|) is a Cauchy sequence.
Transcribed Image Text:2. Let (an) and (yn) be Cauchy sequences of real numbers. Define (rn) to be equivalent to (yn), written (xn) ~ (Yn), if lim n – Yn| = 0. Show that this defines an equivalence relation on the set of all Cauchy sequences of real numbers. Furthermore, prove that if (an) and (Yn) are Cauchy sequences, then ||an – Yn| – |æm – Ym|| < |æn – m| + |Ym – Yn|- Conclude that (|æn – Yn|) is a Cauchy sequence.
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