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

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A:

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Q: 1. Prove by Induction that the numbr f edges of Kn

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A: Set and infimum value

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

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Q: * Let Z be with the discrete topology.11 If A = {1,3,5, 7,...}, then Aº %3D %3D A O N O R O

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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…

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Q: Exercise 6.40. Let E be a nonempty subset of R that is bounded above. Prove that if a = sup E, then…

A: This is the question of topology.

Q: Every nonempty class C has an E-minimal element.

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Q: Prove that Z ≈ E*, where Z is the set of integers and E* is the set of positive even integers.

A: To prove:

Q: If x is a lower bound for S in R then x must be unique..

A: If x is a lower bound for S in R then x must be unique.

Q: Exercise 2. Suppose V and W are finite-dimensional, T E L(V, W), and there exists PE W' such that…

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

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Q: tind the east uprer bound of the set A ntl nE INZ in Q. Prbue that this vaue iS an upper bound and…

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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…

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Q: 2. Let f : X → Y be onto. For each b e Y, let A, {As : b € Y} is a partition of X. f6]. Prove that

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Q: Prove the following. 1. Bolzano-Weierstrass Theorem: If a bounded set S in R" contains infinitely…

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Q: Prove that if a is an upper bound for A, and if a is also anelement of A, then it must be that a =…

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Q: Suppose that f:[a,b] → R is a bounded function for which there sa partition P of [a, b] with L(f,P)…

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Q: Question 2. Prove that a nonempty closed subset of R, if it is bounded from below, has a least…

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Q: Prove that if g′ is bounded on (a, b) then g is uniformly continuous on (a, b).

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Q: of E. 4. Let E CR be a bounded set. Prove sup E e Ë.

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Q: 4.4 Prove that if A is nowhere dense in discrete space (S, d) then A = 0.

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

Q: Let X be an infinite set. Define u : E = P(X) → Ř by %3D if A is finite µ(A) = 1 if A is infinite.…

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Q: |A:k||2||Br:||2) < ||A||F||B||F k=1

A: answer is in next step

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: Let N be with the discrete topology If A=(1,3,5..), then Bd(A)=

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Q: 2. Write a preference E on R as a subset of R', where a bundle a E Rª is at least as good as a…

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Q: Let A C RP and rE RP. Prove that z is a cluster point of A iff there is a sequence of distinct…

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Q: Q Let X be an Infinite sel with topdogy ?. Prove or disprove : (a) If every mfinite Suhsel of x i…

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