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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 Sc(X) CS(X) as the subset of Cauchy sequences in X. Show that So (X)

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 Sc(X) CS(X) as the subset of Cauchy sequences in X. Show that So (X)

Advanced Engineering Mathematics
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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Exercise 4 Part 4 only!
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 (xn)neN E X, i.e. x₂ € X, for all n E N. If (n)neN, (yn)nEN E S(X), define doo (Tn)neN, (yn) nEN) = sup d(xn, yn). nEN 1) 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 map Sconv(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).
Transcribed Image Text: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 (xn)neN E X, i.e. x₂ € X, for all n E N. If (n)neN, (yn)nEN E S(X), define doo (Tn)neN, (yn) nEN) = sup d(xn, yn). nEN 1) 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 map Sconv(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).
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