Let A, B Be two subsets af R, such that B is bounded and YaE A ]bEB such that show that: a sb O A is bounded a bove. sup A < Sup B. prove or disprove that A is bounded below.

Q: Let A be a non-empty subset of R which is bounded below and let B be the subset of R defined by B =…

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

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: Prove that in a metric space (S,d) every closed ball Sr[Xo] is a closed set

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Q: Assume that A and B are nonempty, bounded above, and satisfy B CACR. Show that sup B < sup A.

A: Introduction: The supremum and infimum properties are the very important properties of any subset of…

Q: We define the included point topology by Tp={ UCR;U=Ø or pEU}. Let A = [3,5[, then A is dense in Rif…

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Q: Suppose A ⊂ R is both closed and bounded, prove that A has the Heine-Borel Property

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Q: 2. Let S be a bounded set in R and. let a < 0. and let aS := {as : s E S). Prove that inf(as) = a…

A: Let S be a bounded set in R and let a&lt;0. And let aS := {as : s belongs to S}. We have to prove…

Q: Let A be a non-empty subset ofR which is bounded below and let B be the subset of R defined by B =…

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Q: Let S be a nonempty bounded subset of R and let m = sup S. Prove that m is in S iff m = maxS

A: According to the given information, it is required to prove that:

Q: Let X be an infinite set. (a) Show that is a topology on X, called the finite co со

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Q: Let W be a nonempty subset of R that is bounded above, and define the set B as follows B= {N ER:N is…

A: Let W be a non-empty subset of ℝ is bounded above, and define the set B as follows, B=Ω∈ℝ: Ω is an…

Q: Show that if A and B are bounded subsets of IR, then inf(A U B) = min{inf A.inf B}. %3D

A: To show  If A and B are bounded subsets of R then inf(AUB)=min{infA .infB}

Q: If M is a closed linear subs r space X, and x, be a vect the distance from x, to M, ional f, in X'…

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Q: We define the included point topology by Tp-{UCR;U=Ø or peU). Let A = [3,5[, then A is dense in R if…

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Q: We define the included point topology by Tp={ UCR;U=Ø or pEU}. Let A = [3,5[, then A is dense in R…

A: R is equipped with Tp and p=3

Q: Let A and B be any two subsets of a metric space (X, d). Then (1) A is a closed set. (2) If AC B,…

A: Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: If S is bounded subset of R, then supS E S. Select one: O True O False

A: The given statement is     If is bounded subset of , then .

Q: 2) Let A be a finite set and a EA.Lf a EA is not a maximal element of A, then a has an îmmediate SUC…

A: Given A is a finite set. a∈A is not a maximal element of A. Now we have to prove 'a' has an…

Q: Let A and B be bounded nonempty subsets of R, and let A + B:= {a +b: a E A, be B). Prove that sup(A…

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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: Consider the following subsets of R equipped with the Euclidean topology: A = { neN} Зп — 1 n+1 ° B…

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Q: 6. Let A and B be bounded nonempty subsets of R, and let A+ B:= {a +b: a e A, be B). Prove that…

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Q: Let A be a non-empty set of positive numbers, which is B={re R:1/xe A}. Prove that B is bounded…

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Q: O Suppose that ACR and A is bounded below with inf(A) = L> 0. Prove that for a > 0 and b > 0, the…

A: Since the posted question has multiple questions but according to guidelines we will solve the first…

Q: Let a, be such that asb. Prove that tx Eqt ax cbx

A: Concept: We have a,b∈Q means that a, and b belongs to the rational numbers and a&lt;b means that a…

Q: If A and B are bounded subsets of R, then AuRand AOR are bounded: prove that

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Q: The Borel o-algebra on Ris generated by any of the following collections of intervals 4. {[a, 0) : a…

A: We will use the basic knowledge of set theory to answer this question. We will solve the first two…

Q: Prove that in any metric space (S, d) every closed ball S,[xo] is a closed set.

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Q: If U is homeomorphic to V and U is compact, then * O Vis compact V is complete V is connected O V is…

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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: Suppose the set N is a bounded, open region in R", and that for r > 0, rN := {(rx1,..., rxn) : (x1,…

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Q: Prove that if Gn is an open set which is dense in the complete metric space (S,d) for n = 1,2,...…

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Q: Decide whether the following propositions are true or false.If the claim is valid, supply a short…

A: Consider the statement: Let A be arbitrary, and let K be compact. Then, the intersection A ∩ Kis…

Q: Prove that sup(A) =1, inf(A) = . 5. (i) Let A, BCR be sets which are bounded above, such that AC B.…

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Q: Let M# ¢ is subset of a Hil bert TT H. Show that M is the Smallest Space closed Space. of H which…

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Q: Assume X is Hausdorff. If K C X is compact and x ¢ K, show that there exist disjoint open sets U and…

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Q: For any metric space (X, d) and x EX, the singleton set (x) is closed. Select one: OTrue O False

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Q: Consider a finite set S of points in the two-dimensional plane. A point p of S is called a covidiot,…

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Q: then: Suppose that f:[a,b]→R is bounded , P is parition of [a,b] and Q is refinement of P, L(f;Q) <…

A: It is a mcq type question and not asked about justification so i have provide direct the option.

Q: Consider the following subsets of R equipped with the Euclidean topology: Зп — 1 A = {- n+1 B = Ü 1+…

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Q: f A∩B≠∅ and A∩Bc≠∅ then show that A∩X\B̅≠∅ and A∩B̊≠∅

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Q: Let a,b,c be intergers with gcd(a,b)=1 and c|(a+b) (c divides a+b). Prove that gcd(a,c)=1 and…

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Q: Let A be a non-empty set of positive numbers, which is bounded above and let B = {x e R: 1/x e A}.…

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Q: A metric space is said to be separable if it contains a countable dense subset. (a) Is R separable?…

A: Set of all rational numbers Q belongs to R and Q is dense in R . Also set of all rational numbers Q…

Q: 2) let A be a finite set and a EA.Lf a EA is not a maximal element of A,then a has an îmmediate suc…

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Q: Let R be equipped with the standard topology. Define q : R? → R, x1, if x1 2 x2 X2, if x1 < x2…

A: See the attachment

Q: Suppose that f:[a,b] R is bounded, P is parition of [a,b] and Q is refinement of P, then: L(f;P)…

A: Correct option :- B Detailed Explanation :

Q: Let A be a non-empty set of positive nunıbers, which is bounded above and let B = {x e R: 1/x e A}.…

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Q: Let S be a bounded set in R and. let a > 0, and let aS := {as : s E S}. Prove that: i. inf(as) = a…

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.