3 Let (X, D) Be A Metric Space Define D XxXR By D(X, Y) Min(1, D(X, Y) (A) Prove That D Is A Metric On X (B) Let (Un)Ren Be A Sequence In X Further Let E X Prove That X D Lim , If And Only If X D Lim 100 4 Let X Be The Set Of All Continuous Function (0,5) R Such That F Is Differentiable And F' Is Continuous Define D X X X R By (8,9) 3 Let (X, d) be a metric space Define d XxX R by d(x, y) min(1, d(x, y)) (a) Prove that d is a metric on X (b) Let (a)nen be a sequence in X Further let z X Prove that x d lim x, if and only if x d lim n 117x 7118 4 Let X be the set of all continuous function 0,5 R such that f is differentiable and f' is continuous Define d XxX R by d(f, 9) f(2) g(2) $'(t) g'(t) d dt Prove that d is a metric on X

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Question: 3. Let (X, D) Be A Metric Space. Define D: XxXR By D(X, Y) = Min(1, D(X, Y). (A) Prove That D Is A Metric

3. Let (X, D) Be A Metric Space. Define D: XxXR By D(X, Y) = Min(1, D(X, Y). (A) Prove That D Is A Metric On X. (B) Let (Un)Ren Be A Sequence In X. Further Let : E X. Prove That X = D-Lim , If And Only If X=D- Lim... 100 4. Let X Be The Set Of All Continuous Function : (0,5) ? R Such That F Is Differentiable And F' Is Continuous. Define D: X X X + R By (8,9)
3. Let (X, d) be a metric space. Define d: XxX R

3. Let (X, d) be a metric space. Define d: XxX R by == d(x, y) = min(1, d(x, y)). (a) Prove that d is a metric on X. (b) Let (a)nen be a sequence in X. Further let z X. Prove that x=d- lim x, if and only if x = =d- lim n 117x 7118 4. Let X be the set of all continuous function : [0,5] R such that f is differentiable and f' is continuous. Define d: XxX R by d(f, 9) = f(2) = g(2)| + |* \$'(t) g'(t) \d - dt. Prove that d is a metric on X.

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