Exercise 5Part 1 only! Exercise 5. Let (K, d) be a compact metric space, suppose f: K → K be a map such thatd(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). Showthat d(kp, kq) ≥ e, for all p, q≥ N with p‡q.3) Conclude.

As exercise 5 ,only part 1 is asked to solve; the solution for part 1 only is given below :

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

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

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.

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