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…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.2: Linear Independence, Basis, And Dimension

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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).