Prove that if A is a nonempty set which is bounded below by 3 then B = {x ∈ R | ∃a ∈ A, b = (1/a)} is bounded above by (1/3)

Q: Suppose that A₁, ..., An are bounded subsets of R. Prove that A₁ U… U An is bounded and that sup(A₁…

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Q: Show that if A is such that m (A)> 0, then there exists a bounded subset BCA such that m'(B) > 0.

A: By properties of Measurable set,         If then  .

Q: 60. Let A be a bounded subset of R. Which of these statements is/are not true about A? II: Vx € A,…

A: Option (B) is correct.

Q: Let A and B be bounded NON empty sets of R, and let A + B := { a+b: at A, bt B3. Prove that supla…

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Q: 3. In each case suppose R" has the norm ||l inclusions: = max {ril,., r.}. Prove each of the set (a)…

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Q: Use the Heine-Borel Property from (5.1), which states that a set of real numbers is compact if and…

A: See solution below. Here we used definition of bounded and closed sets.

Q: Find an example of a closed convex set S in R2 such that its profile P is nonempty but convP ≠ S.

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Q: Find an example of a bounded convex set S in R2 such that its profile P is nonempty but conv P ≠ S.

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Q: Let's say that a collection of open balls (Ui)ie1 = (Bx, (ri))iel in Euclidean space Rd is = UB₂,…

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Q: Consider the set T = {..., −34, −17, 0, 17, 34, ... } with the usual definition of less than (a0).…

A: Completeness property of real numbers:  Every non-empty set of real numbers that has an upper bound…

Q: 1. Let X be a nonempty set of sets. Prove that P(UX) = U{P(A)| A e X} if and only if UX E X.

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Q: Q1\ Define the compact set. Then, prove that: if A is a compact subset of F , there exists a best…

A: Compact set : A set is compact if and only if every open cover has a finite subcover. There is also…

Q: Give an example of a set X and topologies T1 and T2 on X such that T1 union T2 is not a topology on…

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Q: 3. Let A = {a} be a bounded subset of R. Which of these three statements about A is'are not true? I:…

A: Given set is A={a} which is bounded subset of R

Q: Find an example of a nonempty set satisfying: 1. Closed, Dense in R 2. Open, Dense in R ~1. & 2.…

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Q: Let S be a nonempty subset of R that is bounded below. Prove that inf S= -sup{-s:s∊S}

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Q: Prove that: m*(A) 2 0, for all sets A

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Q: (a) Suppose S is a subset of positive integers, ZZ. If |S| ≥ 3, prove there exist distinct z and y…

A: (a) Given that S is subset of  such that Observed that any positive integer is either even or odd…

Q: Let X be a non empty set and p be a fixed point in X. Define T={X, ACX such that p€A}. Then any…

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Q: Let A CR be measurable with m(A) > 0. Prove that there is a non-measurable set NCA

A: It is given that A⊂ℝ is measurable and mA>0. The objective is to prove that there is a…

Q: (b) Let F be a non-empty subset of R. Prove that if OF C F, then F is closed.

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Q: 1 Let B be the set of all bounded sequences of real numbers and define the function d: B x B → R by…

A: we have to solve given problem:

Q: Prove that if R: R -> R is continuous, theis b2n the image of a bounded set over f is bounded

A: As per the question we have to prove that if any function f : ℝ → ℝ is continuous, then the image of…

Q: Show that the countable collection B = {(a,b) × (c, d)| a <b and c< d,

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Q: The continuous image of a connected set is connected. That is, if X is connected and f : X–→ Y is…

A: We prove this via contradiction. We assume that f(X) is not connected then X is not connected. And…

Q: 3.6 Prove that a closed set in the metric space (S, d) either is nowhere dense in S or else contains…

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Q: If M and N are two norms on space X then prove that M and N are equivalent if and only if a subset…

A: Any two norms M and N on a norm space is called equivalent if  M≤N≤M. in the given question we have…

Q: 3. Let A = {a} be a bounded subset of R. Which of these three statements about A is/are not true? I:…

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Q: Let A be a nonempty subset of R that is bounded below. Define the set B = {b: b is a lower bound of…

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Q: Given a finite set S, find all possible values of |S] so that the set {(a, b) | (a E S) A (b e S) ^…

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Q: Let Y be an infinite subset of a compact set XC R. Prove Y' 0.

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Q: A relation R defined on the given set A will satisfy

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Q: 4. Assume that S, T C R are two non-empty bounded sets. Prove that: (i) SUT is a bounded set; (ii)…

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Q: Let (X.J) be a topolgical space and {C1, C2 Cn} be a finite collection of compact subsets of X.…

A: Given Let (X,I) be a topological space and C1,C2,......Cn be a finite collection of compact subset…

Q: Prove that he interval E = (a, +oo) is an open set in R. (Show all details of your work).

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Q: Let ACR be a nonempty bounded set. Prove that sup A is unique.

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Q: Let A, B be bounded subsets of R. Let A+B = {r+y:1 € A, y E B} and AB = {ry : 1 € A, y € B}. Which…

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Q: 22. Let A be a bounded subset of R and let sup A = B. Which of the following is/are not an attribute…

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Q: Let T and T 'be two topologies of a set X. Is the family T U T´formed by the openings common to both…

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Q: Let U=[1,2,3,4,5,6,7,8,9], A=[x:x is even and x>=5] B=[2,3,5,7,8,9] Complete the following: (a)…

A: Answer