Sequences and metric spaces Problem 7. Let X be a set. Suppose di and d2 are two different metrics on X and there exists constantsc, k > 0 such thatVæ, y E X : cd1(x, y) < d2(x, y) < k d1 (x, y).cd1 (x, y) < d2(x, y) < k d1 (x, y).Let (xn)1 be a sequence in X. Show that limn→ Xn = x in (X, d1) if and only if limn→ Xn = x in(Х, d2).

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Answered: Problem 7. Let X be a set. Suppose dı…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.CM: Cumulative Review

Problem 17CM

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Problem 7. Let X be a set. Suppose dı and d2 are two different metrics on X and there exists constants c, k > 0 such that Vx, y E X : cdi(x, y) < d2(x, y) < k d1 (x, y). Let (xn)1 be a sequence in X. Show that limn-0 Xn = x in (X, d1) if and only if limn→0 *n = x in (X, d2).