Let A = {(r, y) ER x R; 0< r< 1}, then: O A is not compact because it is not closed in R. None of the choices A is not compact because it is not bounded in R. OA is compact in R.

Q: Let P be a partition on [a,b]. Then ƒ € R[a, b] if and only if inf > sup f.Ax k=1 [xk-1.Xk] sup 2,…

A: To check if the given statement is true or false

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…

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A: Given that,                X is a normed space and a ∈ X . We have to show that there exists an f ∈…

Q: Assume that is a bounded open set in R2 and that g: dR continuous, with the following ball boundary…

A: Given: The given details are, Ω is a bounded open set in ℝ2 and that g : ∂Ω→ℝ continuous with the…

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

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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

Q: Let g be bounded on [a, b] and assume there exists a partition P with L(g,P) = U(g,P). Describe g.…

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Q: f(x,y) = (In(y + x²)) x² + y² D = {(x,y) ERIy2x²-1 and y ≤4-e-*) (b) Observe the set Is D…

A: (b) The given set is D=x,y∈ℝ : y≥x2-1 and y≤4-e-x To find: The given set is bounded or closed or…

Q: Prove the following theorem: If ƒ(x, y) is defined in an open region R of the xy-plane and if ƒx and…

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Q: (a) Let A = { 1.r €R. Show that A is a bounded subset of R. %3D 2x2+1 Hence find inf A and sup A.

A: As per bartleby guidelines for more than one questions asked only first should be answered. Please…

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}.…

A: We have, A=set of positive numbers  A is bounded above set so, for all  x∈A ∃ K∈ℝ such that x≤K We…

Q: Let x be O and all subsels f X whose complements Prove that (X, z) infimite set and let z consist of…

A: Consider, I=A⊂X : Ac is countable ∪ϕSince,Xc=ϕ is countable, X∈I Also given, ϕ∈I Assume, Aii∈N∈I…

Q: Let A = { (x,y)e RXR; 0<max{}x-31. ly-21)<1}, then: %3D None of the choices OA is not compact…

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Q: Let P, be an equidistant partition of [2, 4] on f(x) = x , then U(f, P,) =

A: To find the U(f, pn) of the function f(x) = x

Q: E be a Euclidean space of dimension (E(1, f)) = dim(Ker (f- id)), then s minimal.

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Q: Show that R is not compact.

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Q: prove that an is bounded above.

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Q: Set (Q, M, µ) a space of o-finite measure. we define the aplication 4: L° (Q) → L'(Q)* given by =…

A: Let g,h∈L∞(Ω and α∈F Claim:- ϕ(αg+h)=αφ(g)+φ(h)

Q: Let A= {(x, y) E R × R; 0 < 1 < 1}, then: O A is compact in R. A is not compact because it is not…

A: A set K is said to be compact set if every open cover C has a finite subcover M which covers set K.

Q: Let X = R, and d be the diserete metrie, and A= {1,2) Show that if A is compact or not

A: Given set A={1,2}

Q: Suppose (X,S,u) is a measure space and f,h: X → F are S-measurable Prove that I|fh||, < ||f||, ||4|a…

A: Given X,S,μ is a measure space and the functions f,h:X→F are S-measurable. Suppose 1p+1q=1r,…

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: Let (x,T) be a topologīcal space, K a compact subset of X and (FilneNJ famíly in P(X) with Fn+Ø for…

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Q: Let f E R[a, b] (which implies that f is bounded). 1. For any partition of a, b], show that U (2f,…

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Q: If f has an essential singularity at zo and is holomorphic in a punctured neighbourhood U := Br(zo)…

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Q: an open ball in the set X, with its radius & satisfying -d(x, a), Vx € X at the open ball Be (x)…

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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: (d) Consider the set [1, √5] C [−1, √10]. Explain f is not continuous by connectedness, compactness,…

A: Given: The function, f:-1,10→ℝ defined by, fx=x2,x&gt;2x,x≤2 The set 1,5⊂-1,10. To explain: Why the…

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

A: Given that f:X→Y is an onto function. Then for every b∈Y there exists x∈X such that  fx=y Also,…

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

A:

Q: Suppose the collection = {B₁ (x): xe (0,1)}. Is B an open cover of the interval [0, 1) CR? Give an…

A: Given: The collection B=B12x : x∈[0, 1). To determine: Whether B is an open covering of the interval…

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

A: Given the statement Suppose that f: [a, b] → R is a bounded function for which there is a partition…

Q: Suppose f(x, y) is continuous and non-negative over a non-empty open region R in the plane with a…

A: Based on given data,For Non empty open region R,fx, y is continuous,andfx, y is non -…

Q: Prove the following statements for arbitrary elements x, y, z of an ordered integral domainD.   .If…

A: x&gt;y and z&gt;w , then x+z &gt; y+w in an integral domain D

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: Let A = {(r, y) ERx R; 0 <r<1}, then: O A is compact in R. O A is not compact because it is not…

A:

Q: If f(z) = u + iv is entire and u(x, y) is bounded above, apply Liouville's Theorem to exp(f(z)) to…

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Q: (b) Let # ACR be bounded below. Prove that Inf (A) =-Sup(-A). %3D

A: Let A is non-empty subset of real number . Let A be bounded below then we have to show that  Inf(A)…

Q: The net of a cone consists of three different two-dimensional areas. O True O False

A:

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: Let A be compact and U C A be closed. Prove that U is compact.

A: The complete proof is in given below

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

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Q: (b) Given two metric spaces X,p>, and a function f : X→ Y that is uniformly continuous on SC X. If…

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Q: 7. Assume that f:XY is a function from one metric space (X,d,) to another(Y,d,). Prove that f is…

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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}.…

A:

Q: . Suppose f(a, y) is continuous and non-negative over a non-empty open region R in the plane with a…

A: According to question,f(x, y) is continuous,fx, y ≥ 0        non - negative,∫∫Rfx, y dA = 0

Q: Consider the set F=(-infinity, 1) U (9,infinity) as a subset of R with d(x,y)=abs(x-y). Show f is…

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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}.…

A:

Q: 1. Let X C R"-1 be bounded. Prove that X considered as a subset of R" is a volume zero set.

A: Given, X⊆ℝn-1 That is, X is a subset of ℝ in terms of exponential n. So, let we consider the set,…