Exercise 4. Let (X, d) be a metric space such that d(x,x) ≤ 1, for all x, x' X. Denote by S(X) the set of sequences (Tn)neN E X, i.e. In € X, for all n € N. If (xn)neN; (Yn)nEN E S(X), define do (Tn)neN, (Yn)neN) = sup d(xn, Yn). nEN 1) Show that do is a distance on S(X). In the sequel, we endow S(X) with the topology of the metric do 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 map Sconv(X) → X, (Zn)n€N →limn→+o ®n is Lipschitz. 5) Define Scauchy (X) CS(X) as the subset of Cauchy sequences in X. Show that Scauchy (X) is a closed subsset in S(X).
Exercise 4. Let (X, d) be a metric space such that d(x,x) ≤ 1, for all x, x' X. Denote by S(X) the set of sequences (Tn)neN E X, i.e. In € X, for all n € N. If (xn)neN; (Yn)nEN E S(X), define do (Tn)neN, (Yn)neN) = sup d(xn, Yn). nEN 1) Show that do is a distance on S(X). In the sequel, we endow S(X) with the topology of the metric do 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 map Sconv(X) → X, (Zn)n€N →limn→+o ®n is Lipschitz. 5) Define Scauchy (X) CS(X) as the subset of Cauchy sequences in X. Show that Scauchy (X) is a closed subsset in S(X).
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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