• Problem 8. Let || . || be any norm on R". Let x,y: (a,b) + R" be continuous and let so € (a,b) be such that x($0) = y(

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Problem 8 Let Be Any Norm On R Let X Y A B R Be Continuous And Let So A B Be Such That X 0 Y 1 (95.41 KiB) Viewed 41 times

• Problem 8. Let || . || be any norm on R". Let x,y: (a,b) + R" be continuous and let so € (a,b) be such that x($0) = y(so). We assume the following statement, which we label Assumption (A): "if there exists an s E (a,b) so that x(s) = y(s), then there exists some 8 >0 so that for all te (s - 0,8 + 8) we have x(t) = y(t)." Suppose there exists 81 € ($0, b) so that x(81) + y(81). (1) Show that the set E := {t E (a,b) : (t) = y(t) and t < $1} is not empty. (2) Show that the set E is bounded above by $1. (3) Show that T := sup E is such that T E (a,b). (4) Use the assumption (A) to show, if T E E, then there exists some SEE with T < S so that for all te [T, S) we have e(t) = y(t). Explain why this contradicts the definition of T as the supremum of E. Conclude that we cannot have TEE.