below. If y = inf S, show that, for each delta> 0, there is s ∈ S such that y ≤ s < y + delta .

Q: Take any r > 0 and any c ∈ R. Show that the r-ball Br( c ) is an open subset of R.

A: We need to prove that for any r &gt; 0 and any c ∈ ℝ  the r-ball Br( c ) is an open subset of ℝ.

Q: 5. Let E be a set that bounded and c > 0. Prove that sup(cE) = -c inf(-E) %3D

A: This is a problem of Real topology, metric topology.

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A: Two sets which are one-one and onto, have same cardinality. Let S is a countable set,  S=a1,…

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A:

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Q: 2. Consider the set A = { n+m² n2+m m eN}. : п, тEN (a) Prove that sup(A) = ∞. (b) Prove that A is…

A: Set and infimum value

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A: We have to show that the given set S is a denumerable set.

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A:

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}. 1…

A:

Q: Let S be a non-empty bounded subset of R and let a be an upper bound of S. Show hat sup S = a if and…

A: Follow the steps.

Q: 21. Let A and B be two non-empty bounded sets of positive numbers and let C = {xy : x e A and y e…

A:

Q: . Let S be a bounded set in R and let T be a nonempty subset of S. Show that inf S <inf T< sup T <…

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Q: 23. Let A be a non-empty set of positive numbers, which is bounded above and let B = {xe R: 1/xe A}.…

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Q: Let M and N be nonempty bounded subsets of R and let M + N := {m + n : m ∈ M and n ∈ N} . Prove that…

A:

Q: is not an endpoint of any of the intervals (e) Prove that eC but in any of the sets A (k e N),

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Q: Given a metric spaceX,p> (a) If the sequence (2n)neN C X is convergent, show that it is bounded.

A: Given: Metric space X,ρ. To Prove: In a metric space X,ρ, if a sequence xnn∈ℕ⊂X is convergent then…

Q: Q2: Let the set under consideration be N for each n EN, define u, = {n, n+1, n+2, ...} and let 1 =…

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Q: If A is bounded below and B is bounded above, show that inf(A − B) = inf A − sup B

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Q: be any nonempty set and let µ* be an outer measure on P(X) and let A, B e P(X). If at ne of A, B is…

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Q: Let N be a measurable set of finite measure and let 1 <p< q < ∞∞. Show that if fn f in Lª(N), then…

A:

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

A:

Q: Consider the subset T = {r €R: , <0} of R. x² –3x 3.1 Express T in interval notation. 3.2 Choose ONE…

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Q: Show that every open set with respect to 7 is a union of open ball(s). A point p is a limit point of…

A: Given X be a matric space with metric d. For any r&gt;0, the open ball at center at x∈X  with radius…

Q: Find the image of the given set under the exponential mapping The semi-infinite strip ∞o < x≤0,0 ≤ y…

A: Given That : Exponential mapping  the semi-infinite stripe ∞&lt;x≤0, 0≤y≤π To Find :Image the given…

Q: . Let E be a measurable set with m(F np with ISp <∞. Further if Lim (E) then lI f llo = %3D

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

A:

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A: Consider A and B are bounded sets.

Q: Show that Let ECR" is measurable set. If µ(E) >0, then E have a non-measurable subset

A: Here, we introduce the notation Gr :=G∩[0, r) + 1-r ∪ G∩[r,1) - r for G⊂[0,1) and r∈R. Note that…

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Q: The set S=xER:x²-4<0} with the usual metric is . O A. Compact. B. connected. C. Not connected D.…

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Q: Let E and F be measurable sets each having finite measure. Prove that m(EAF) = 0 if and only if m(E)…

A: To prove: The statement "Let E and F be measurable sets each having finite measure. Then, mE∆F=0 if…

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Q: 16. If S cT c R, where S # $ , then show that (i) If T is bounded above, then sup (ii) If T is…

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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: Suppose u is a finite Borel measure on R, which is singular to Lebesgue measure. Show that except…

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Q: 23. Let A be a non-empty set of positive numbers, which is ounded above and let B= {xe R: 1/x e A}.…

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Q: X be a limit point of set ECR, then every o infinitely many points of terval containing

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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: Define u* on the power set of R by Jo if A is bounded, 4*(A) = 1 if A is unbounded. Is u* an outer…

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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: 5. Let S be a nonempty subset of R that is bounded above, with upper bound b. Prove that b = sup(S)…

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Q: . A finitely additive measure u is a measure iff it is continuous from below as in meorem 1.8c. If…

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Q: Q1- If f is bounded and define on [a, b] then show that m(b – a) < L(P,f)< U(P,f) < M(b – a). Q2-…

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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…

A: