(b) Prove that R is numerically equivalent to any bounded open or closed interval.
Problem 3. Read about compactness in Section 2.8 of the book. Then, prove, WITHOUT RELYING ON HEINE-BOREL's THEOREM, the following. Let E be a closed bounded subset of E and r be any function mapping E to (0,00). Then there ensts finitely many pints yi E E,i = 1, , N such that i-1 Here Br(y.)(y) is the open ball (neighborhood) of Tudius r(y.) centered at yi. Problem 3. Read about compactness in Section 2.8 of the book. Then, prove,...
Prove that in R^n with the usual topology, if a set is closed and bounded then it is compact.
(a) Suppose f is continuously differentiable on the closed and bounded interval I = [0, 1]. Show that f is uniformly continuous on I. (b) Suppose g is continuously differentiable on the open interval J = (0,1). Give and example of such a function which is NOT uniformly continuous on J, and prove your answer.
Exercise 5.3.2. [Used in Exercise 5.5.6.] Let [a,b] C R be a non-degenerate closed bounded interval, and let f: la,b] R be a function. Suppose that f is integrable Prove that if If(x)l S M for all xe la, b], for some M E R, then Jx)ds M(b-a) Exercise 5.3.2. [Used in Exercise 5.5.6.] Let [a,b] C R be a non-degenerate closed bounded interval, and let f: la,b] R be a function. Suppose that f is integrable Prove that if...
Let R be an interval (open, closed, neither are all fine) and let f: I-> R be a continuous strictly increasing function. Do the following: (1) Show that the inverse function f -1 exists. (2) Prove that f is an open map (in the relative topology on I) (3) Prove that f1 is continuous Let R be an interval (open, closed, neither are all fine) and let f: I-> R be a continuous strictly increasing function. Do the following: (1)...
please do not use “compact” term, i have not covered that in class Prove that the open interval (-π/2, π/2), considered as a subspace of the real number system, is topologically equivalent to the real number system. Prove that any two open intervals, considered as subspaces of the real number system, are topologically equivalent. Prove that any open interval, considered as a subspace of the real number system, is topologically equivalent to the real number system Prove that the open...
2. In the below, by "inite open interva" we mean an interval (a, b) where a,b e R and a < b. And by finite closed interval" we mean an interval [a, b] where a, b e R and a<b. (a) Let f : A → B be a continuous function where f(A) = B. Is it possible for A to be a finite open interval while B is a finite closed interval? Either provide an example showing it is...
1. Prove that for any set S S R, S is closed if and only if Se is open. Notice the book has a proof of this, but it uses a different notation for set complements and a different definition of neighborhood. You may consult it, but you must write your proof using the definition for interior point I presented in class (also in the notes on blackboard). If you copy the proof from the book you will not receive...
(4) Let (Ω,A) be a measurable space, and let f : Ω → R. Prove that the following statements are equivalent: ·f is measurable. ·f-1(1) E A for any open interval I c R. lei f (A) E A for any open set ACR ·f-1 (A) E A for any Borel set A c R. (4) Let (Ω,A) be a measurable space, and let f : Ω → R. Prove that the following statements are equivalent: ·f is measurable. ·f-1(1)...
(4) Let (Q,A) be a measurable space, and let f : Ω-> R. Prove that the following statements are equivalent: f is measurable . f-(I) E A for any open interval I CR .f-(A) E A for any open set ACR. . f-(A) E A for any Borel set ACR. (4) Let (Q,A) be a measurable space, and let f : Ω-> R. Prove that the following statements are equivalent: f is measurable . f-(I) E A for any open...