Let A1, A2, ., Am be finitely many subsets of R'. Assume that z is a cluster point of UA;. Prove that there exists some 1 < j < m such that z is a j=1 cluster point of the single set A,.

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: If A and B are bounded subsets of R, then prove that AUB and AB are bounded.

A:

Q: Let A be a nonempty subset of R. If a = sup A is finite, show that for each ɛ > 0, there is an a E A…

A: We have to prove the given statement

Q: Consider the collection of all points of the form – 2w] 3w w Express this collection as the span of…

A:

Q: Prove: If S is compact, and ƒ is continuous on S, then ƒ takes a minimum value some- where in S.

A:

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

A:

Q: e) Prove the theorem that states that “S" is a countable set, and there exist a surjection of N onto…

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

Q: Show that every countable subset of R" has Hausdorff dimension 0.

A: claim- to show that every countable subset of ℝn has Hausdorff dimension 0. it is known that…

Q: Suppose that A and B are bounded, nonempty subsets of R. Prove that AU B is bounded and that sup(AU…

A: Consider the provided question, Suppose A and B are bounded, nonempty subsets of R.

Q: Prove that the set S = {n² : n e N} is denumerable. %3D

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

Q: 2. Let S is a subset of R. Prove that S is closed if and only if S contains all of its accumula-…

A:

Q: Show that the set (1) isopen u k = 1 and find its measure.

A:

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

A:

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

A: To prove:

Q: Show that there exists a set N C (0, 1) that is not Lebesgue measurable.

A:

Q: Give an example of a closed subset S of R^2 such that conv S is not closed

A: An example of a closed subset S of R2 such that conv S is not closed is given below.

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…

A:

Q: Show that for any space A there does not exist an injection P(A) → A.

A: To prove that for any space A, there doesn't exist any injection PA→A Let us consider the contrary.…

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 x E RP be a cluster point of a set E and that x ¢E. Prove that x is a boundary point of E.

A:

Q: Let {An : n e N} be the indexed collection of sets defined by (2. 2-) 1 An Prove that: = Ø

A:

Q: Let A be the subset of R defined by A := {1-Q" :n eN}. Find all cluster points of A and justify your…

A:

Q: Prove that every finite set S of R" has no accumulation point.

A: Now taking set S Let ? = {?1, ?2, . . . , ?n }.

Q: If A and B are nonempty bounded subsets of R with A C B, prove that sup A < sup B.

A:

Q: Let A be a nonempty set and define B={kx : x∈A}, where k≥0 is fixed. Prove that supB=ksupA.

A: Given A be a nonempty set and define B=kx:x∈A where k≥0 is fixed. We have to prove that supB=ksupA…

Q: Give an example of a closed subset S of R2 such that conv S is not closed.

A:

Q: 3. Prove that the image of a measurable set E under the translation 4 is measurable and m(E+a)…

A:

Q: Let Xbe a nonempty subset of R. Show that if c E R is a cluster point of X, then for any n EN, there…

A:

Q: Let (X. T) be a T, topological space and A a nonempty subset of X, if x is a cluster point of A…

A: Given: Let (X,T) be a T1 topological space and A a non-empty subset of X, if x is a cluster point of…

Q: 2 Let A be a fixed nonempty subset of set X and I= {Bc X | A Z B}. Show that (X,I) %3D is a matroid.

A:

Q: Let (S,≼) be a linearly ordered set with the Least-Upper-Bound Property. Let A and B be non-empty…

A:

Q: If A = (0, 1) n Q, then {x; x E (0, 1)} is the set of all cluster points of A. Select one: OTrue…

A: Given A= (0,1) ∩ Q . Then the set of all cluster points of A is  = {x: x ∈ [0,1]}

Q: 6. If A has m members and B has n members, prove that A x B has mn members 7. Sketch each of the…

A:

Q: 11 Prove that if A CR and |A| > 0, then there exists a subset of A that is not Lebesgue measurable.

A: As per the policy, we are solving a first question. Please repost the question and specify which…

Q: Let A be a nonempty subset of a bounded set B. Why does inf A and sup A exist? Show that (a) inf B ≤…

A:

Q: (b) Prove that the complement of a dense open subset of R is nowhere dense in R..

A: Prove that the complement of a dense open subset of R is nowhere dense in R.

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.…

A: Let X be an infinite set and ∑ =P(X) Let μ=P(X)=∑ → ℝ is given by μ(A)=0if A is finite1if A is…

Q: Show that there must be two distinct subsets A, BC S such that

A: Every value in S is between 0 and 100 , So the sum of any subset will be at least 0 and  at most…

Q: Consider the subset G {r• /2: x € Z} of R. Show that G is closed un- der addition.

A: The given subset is: G= x·2 : x∈ℤ

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:

Q: Let A be a subset of R and let c€ R. Explain what it means to say that c is a cluster point of A.

A:

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

A:

Q: Find an infinite collection {Kn: n N} of compact sets in R such that the union U K₂ is not compact.…

A: We know the definition of compact sets , A subset S of ℝ is said to be compact if every open cover…

Q: Prove that for every natural number v ≥ 4 there exists a planar graph with v vertices which has all…

A: Let v∈ℕ such that v≥4. Therefore there exists a natural number k such that v=4k+r, where 0≤r&lt;4.…

Q: Let A = {x E Z* : x is a mutiple of 2 and x < 15}. Rewrite A in roster notatlon.

A: Given that : A = {x∈Z+ : x is a multiple of 2 and x &lt;15 }   Roster form : A representation of a…

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

A: