© 93 1 8:01First Quiz-FA-414-14-4-2021.pdfالمادة :مدرسQuestion 1 : If d, and d, are metrics on the same set X and there are positive integers a and b such thatfor all x, y E X,ad (x, y) < d>(x, y) < bd (x, y).Show that the Cauchy sequences in (X, d1) and (X, d2) are the same.Question 2 : Show that the set of all real numbers constitutes an incomplete metric space if we choosed(x, y) = |tan-'x – tan'y|.Question 3 : If X and Y are isometric and X is complete, show that Y is complete.d(x, y)1+ d(x, y)Question 4 : Let (X,d) be a metric space and let p(x, y) =for all x, y E X. Show that (X, d)is complete if and only if (X, p) is complete.Question 5 : What is the completion of (X, d). where X is the set of all rational numbers and d(x, y)|r – y| ?

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Description: © 93 1 8:01First Quiz-FA-414-14-4-2021.pdfالمادة :مدرسQuestion 1 : If d, and d, are metrics on the same set X and there are positive integers a and b such thatfor all x, y E X,ad (x, y) (x, y) < bd (x, y).Show that the Cauchy sequences in (X, d1

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Question 1 : If d, and d, are metrics on the same set X and there are positive integers a and b such that for all x, y E X, ad, (x, y) < d>(x, y) < bd1 (x, y). Show that the Cauchy sequences in (X,d1) and (X, d2) are the same.

Question 1 : If d, and d, are metrics on the same set X and there are positive integers a and b such that for all x, y E X, ad, (x, y) < d>(x, y) < bd1 (x, y). Show that the Cauchy sequences in (X,d1) and (X, d2) are the same.

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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