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Exercise 5. Let (K, d) be a compact metric space, suppose f: K→ K be a map such that d(f(x), f(y)) = d(x, y), for all x, y ɛ K. The goal is to show that f(K) = K. Assume by contradiction that ko ‡ f(K).

Exercise 5. Let (K, d) be a compact metric space, suppose f: K→ K be a map such that d(f(x), f(y)) = d(x, y), for all x, y ɛ K. The goal is to show that f(K) = K. Assume by contradiction that ko ‡ f(K).

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 5 Part 1 only!
Exercise 5. Let (K, d) be a compact metric space, suppose f: K → K be a map such that d(f(x), f(y)) = d(x, y), for all x, y ɛK. The goal is to show that f(K) = K. Assume by contradiction that ko & f(K). 1) Show that there exists € > 0, such that B(ko, €) ƒ(K) = Ø. 2) Define kn, n N by induction on n. Start with kŋ, for n ≥ 1, define kn+1 = f(kn). Show that d(kp, kq) ≥ e, for all p, q≥ N with p‡q. 3) Conclude.
Transcribed Image Text:Exercise 5. Let (K, d) be a compact metric space, suppose f: K → K be a map such that d(f(x), f(y)) = d(x, y), for all x, y ɛK. The goal is to show that f(K) = K. Assume by contradiction that ko & f(K). 1) Show that there exists € > 0, such that B(ko, €) ƒ(K) = Ø. 2) Define kn, n N by induction on n. Start with kŋ, for n ≥ 1, define kn+1 = f(kn). Show that d(kp, kq) ≥ e, for all p, q≥ N with p‡q. 3) Conclude.
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