Exercise 4Part 4 only! Exercise 4. Let (X, d) be a metric space such thatd(x, x') ≤ 1, for all x, x' X.Denote by S(X) the set of sequences (xn)neN E X, i.e. x₂ € X, for all n E N. If (n)neN, (yn)nEN ES(X), definedoo (Tn)neN, (yn) nEN) = sup d(xn, yn).nEN1) Show that doo is a distance on S(X).In the sequel, we endow S(X) with the topology of the metric d.2) If (X, d) is a complete metric space show that (S(X), d) is a complete metric space.3) If (S(X), d∞) is a complete metric space show that (X, d) is a complete metric space.4) Define Sconv (X) C S(X) as the subset of convergent sequences in X. Show that the mapSconv(X) → X, (2n)neN → limn→+oo ^n is Lipschitz.5) Define Scauchy (X) C S(X) as the subset of Cauchy sequences in X. Show that Scauchy (X)is a closed subsset in S(X).

Answered: Image /qna-images/answer/bdef4211-b141-4154-bb26-cca61b2c15fc.jpg

Answered: 3) If (S(X), d) is a complete metric…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.4: Ordered Integral Domains

Problem 8E: If x and y are elements of an ordered integral domain D, prove the following inequalities. a....

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If x and y are elements of an ordered integral domain D, prove the following inequalities. a. x22xy+y20 b. x2+y2xy c. x2+y2xy
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