7. Suppose f is continuous at x. a. Suppose f(x)>0. Prove that for x close...
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7. Suppose f is continuous at x. a. Suppose f(x)>0. Prove that for x close enough to x,f(x)>0. b. Prove that xix,f(xi)>0 imply f(x)0. c. Give an example in which xix,f(xi)>0 but f(x)=0. 8. Suppose that X is a non-empty set with a metric d defined on it. Prove that M, a subset of X is closed if and only if {xi}M and xix imply xM
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7. Suppose f is continuous at x. a. Suppose f(x)>0. Prove that for x close enough to x,f(x)>0. b. Prove that xix,f(xi)>0 imply f(x)0. c. Give an example in which xix,f(xi)>0 but f(x)=0. 8. Suppose that X is a non-empty set with a metric d defined on it. Prove that M, a subset of X is closed if and only if {xi}M and xix imply xM 