If S and T are nonempty bounded subsets of the real numbers, show that if ScT Then infT < inf S < sup S < sup T

Q: If S is a non-empty set of real numbers which is bounded above, then a real number s is the supremum…

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Q: Suppose that A and B are subsets of R which are nonempty and bounded above, and suppose that BCA.…

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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: Let A be a nonempty subset of positive integers and B = {n + v5 : n E A}. Show that B has a least…

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Q: (c) Let us assume the facts from part (b) for all a, b e R, i.e. as < bs whenever a < b. Let SCR be…

A: Given that S~=x∈ℝ|x3∈S And S⊂ℝ is bounded above. We say that S⊂ℝ is bounded above if  and if…

Q: If S is a non-empty subset of R which is bounded below, then a real number t is the infimum of S iff…

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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}. 1…

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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: 3) Let A be a nonempty set of real numbers that is bounded below. Let -A be the set of all numbers…

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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}. 1…

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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 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: (a) Prove that if A and B are nonempty, bounded sets of real numbers such that x ≤ y for all x € A…

A: a) It is given that A and B are nonempty, bounded sets of real numbers such that x≤y for all x∈A and…

Q: . Let A CQ satisfying the following conditions: (a) A + Ø and A # Q. (b) If p E A and q E Q with q <…

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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 S be an ordered set.  Let B ⊂ S be bounded (above and below). Let A ⊂ B be a nonempty subset.…

A: We can prove this by the definition of upper and lower bound

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

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Q: Prove that if ø is convex, then a-[$(x + Az) – Þ(x)] is nondecreasing for å > 0.

A: As  φ is convex we conclude that its second derivative :  φ'' is always positive ∴  φ''(x) &gt; 0…

Q: If S is a non-empty subset of R which is bounded below, then a real number t is the infimuni of S…

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Q: If X has full rank, so that P = X(X'X)-'X', prove that C(P) = C(X).

A: Given information: It is given that X is a full rank matrix.

Q: Suppose that fa f (uniformly) and gn= g (uniformly) (pointwise) on A CR. on a set ACR. Show that…

A: “Since you have asked multiple question, we will solve the first question for you. If you want any…

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…

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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: 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: Suppose that |x| = n. Find a necessary and sufficient condition on r and s such that (x") C (x*).

A: Given,

Q: Consider the subset W = {x € Q : xª > 1} of Q. 2.1 Express W using interval notation. 2.2 Is W…

A: W = x∈ℚ: x4&gt;1 This implies that W is a set of rational numbers 'x', such that x4&gt;1. if x4…

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: 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: 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: IF f:x-у and vCy proof fy/v)=x/f"(v) and proof f is continous iff f'(E) is closed set in x for every…

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Q: Suppose that S is a nonempty subset of R and k is an upper bound of S. The number k = sup S if and…

A: Given: S is a non-empty subset of R and k is an upper bound of S. To prove : Number k=sup S if and…

Q: For any measurable nonempty subset of R, L∞ (N) = I¹ (N).

A: Given: Measurable nonempty subset Ω of ℝ. To prove: For any measurable nonempty subset Ω of ℝ,…

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

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Q: if f ofen mapp Prove that for every. xeX and for every ofen Set Vxe Tx in x contains the Point x…

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Q: Which characteristic/s implies(y) that a set S of real numbers is closed? S has no accumulation…

A: To choose correct option from given question.

Q: Let F and G be nonempty bounded sets. Prove that inf (F UG) = min{inf F, inf G} and sup (F UG) =…

A: Since F and G are non-empty bounded subsets of R, sup F, sup G, inf F and inf G are all exist in R.…

Q: 3. Let E be a nonempty subset of R that is bounded above, and set U = {ß ER is an upper bound of E}.…

A: Given that Let E be a nonempty subset of R that is bounded above, and set U = { B belongs to R is an…

Q: Let A be nonempty bounded subset of real numbers, with u=sup{A}>0. Let A= for all a = 0}, then inf(-…

A: Since you have asked multiple questions in a single request so we will be answering only first…

Q: .If o + S CR is bounded above, then S has a supremum

A: It is given that S is a non-empty sub-set of R and R is bounded above.

Q: Let x= sup (A) and if a E A, with x# a then which of the following statements is true? There is rEA…

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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: Suppose A and B are non-empty sets of real mumbers that are both bounded above. (a) Prove that, if A…

A: (a)

Q: If S is a non-empty subset of R which is bounded below, then a real number t is the infimunm of S…

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Q: Let A be a nonempty subset of positive integers and B = {n + v5_ : nE A}. Show that B has a least…

A: We will use the well ordering principle to prove that B has a least element.

Q: Suppose A and B are non-empty bounded subsets of positive real numbers. Define A·B = {r •y:x € A, y…

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Q: If S is a non-empty subset of R which is bounded below, then a real number t is the infimuni of S…

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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: Suppose A C R and |A| 0 there exists a set G that is the union of finitely many disjoint bounded…

A: suppose that A⊂ℝ and A&lt;∞ To prove : A is lebesgue measurable iff for all ∈&gt;0 there exist a set…

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.