Complete the proof by filling in the blanks Proposition 4.3.14. Let (S, d) be a metric space. Then for any a and e> 0, B(x, e) is open. Proof. To show that B(x, €) is open, we must show that, for anyy e B(x, e) there exists a ô > 0 (we have to use d here since e is already defined) such that B(y, 5) C1. If this is true, then y is in the 2 of B(x, E), which would show B(x, €) C Int (B(x, E)), since y was chosen arbitrarily. Notice that d(r, y) 0. Let 5 =€ -d(x, y). Now we will show that this choice of & gives us B(y, 5) C B(x, € ). To show B(y, 8) C B(x, E), suppose ze B(y, 8). By the triangle inequality of metrics, we have d(x, z) < d(x, y) + 3< d(x,y) + 6 = d(x, y)+ € -d(x, y) = [4

Complete the proof by filling in the blanks Proposition 4.3.14. Let (S, d) be a metric space. Then for any a and e> 0, B(x, e) is open. Proof. To show that B(x, €) is open, we must show that, for anyy e B(x, e) there exists a ô > 0 (we have to use d here since e is already defined) such that B(y, 5) C1. If this is true, then y is in the 2 of B(x, E), which would show B(x, €) C Int (B(x, E)), since y was chosen arbitrarily. Notice that d(r, y) 0. Let 5 =€ -d(x, y). Now we will show that this choice of & gives us B(y, 5) C B(x, € ). To show B(y, 8) C B(x, E), suppose ze B(y, 8). By the triangle inequality of metrics, we have d(x, z) < d(x, y) + 3< d(x,y) + 6 = d(x, y)+ € -d(x, y) = [4

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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