1. Suppose that {an} and {b.) are bounded sequences. Show that lim sup an lim supbn. lim sup an +bn 2. Show that R with the Euclidean metric is the union of countably many bounded sets. Is that true in any metric space?

Q: 8. If A is not totally bounded, show that A has an infinite subset B that is homeo- morphic to a…

A: Infinite subset can be termed as the set of natural numbers have the subset of its number in the…

Q: Consider the space Z+ with the finite complement topology. Consider the sequence (xn) of points in…

A: (ℤ , +) with finite complement topology. (ℤ+ , τC)   where τC={U⊆ℤ+ |ℤ+ ∖U is finite } The sequence…

Q: Find the Limit of the Sequnce 2a, § at n-sa∞ or prove that in the limat does not exist An= Un²_ 4n+…

A: Take an  and solve for n tends to infinity to find the limit

Q: If F is a closed bounded set, then every infinite subset S of F has a limit point in F.

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Q: Give an example of a metric space X, d and a sequence {Pn} C X such that {pn} is a Cauchy sequence…

A: Let X=(0,∞) Define d:X×X→ℝ+∪0 by d(x,y)=x-y.i.e. d is a standard metric on X induced from ℝ.Take…

Q: Show that lim,--zo f (z)g(z) = 0 if limz¬zo f (z) = 0 and if there exist a positive number M such…

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Q: Let c be a cluster point of a subset A of R and let f : A → R be a function. Suppose for every…

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Q: Show that a linear map T:X-→ Y between normed spaces is continuous if and only if {Tx,} is a Cauchy…

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Q: Compute the following limits of sequences. (n2")? lim no + n² ln(n) + n²3n´

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Q: Let (X, d) be a metric space and (r.) is a sequence in X. (a) What is it meant by saying (zn)…

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Q: 6. Show that the following sequences of sets, {C& }, are non-increasing, (1.2.17), then find lim-»∞…

A: * SOLUTION :-  (6) Given sequence of sets , {CK}, non- increasing, (1.2.17),

Q: 2.1 Let B be the set of all bounded sequences of real numbers and define the function d : B × B → R…

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Q: In formally proving that lim (2² + x) (유, 1). = 42, let e > 0 be arbitrary. Choose 8 = min Determine…

A: Given limx→6x2+x=42 Consider x2+x−42=x2+7x−6x−42=xx+7−6x+7=x−6x+7≤x−6x+7 Hence, x2+x−42≤x−6x+7.....1…

Q: Prove the following analogue of Theorem 3.10(b): If {Ein} is a sequence of closed nonempty and…

A: given,En is a sequence of closed nonempty and bounded sets in a complete metric space X. En⊃En+1 and…

Q: 5. Prove: Let A₁, A2, be measureable sets such that limn→∞ µ(An) Im C = 0. Then for every

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Q: Let c be a cluster point of a subset A of R and let f : A → R be a function. Suppose for every…

A: Consider the provided question, Let c be a cluster point of a subset A of R and let  f : A → R be a…

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: Let X be a set, let fn : X → R be a sequence of functions, and let f : X → Y be a function. Let ē…

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Q: In(7)/ In(x) (17 + 8x?) Find lim Limit is equal to (Type an integer.)

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Q: Suppose that a, > 0 and lim a, = ∞. Prove that Sa, diverges.

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Q: if {an} is not bounded above, then lim sup an = 00 صواب ihi

A: We Know that  A sequence is bounded above if all its terms are less than or equal to a number K',…

Q: 4.1 Prove that in any metric space (S, d) every closed ball S,[xo] is a closed set. 4.2 Let F be a…

A: Given (S,d) is a metric space. Three questions related to this are given. The solution to these…

Q: Find the horizontal limit(s) of the follov

A: Solution of the problem as follows

Q: Suppose that C is a cluster point of A and f:A → R. To say that lim f(x) = Lis equivalent to saying…

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Q: ã) In general, for any metric space (M, d), we have 1) Sis compact in M = S is closed and bounded in…

A: Our guidelines we are supposed to answer only one question. Kindly repost other question as the next…

Q: 8. If (X, M, ) is a measure space and {E₁} CM, then (lim inf E₁) lim inf μ(E). Also, μ(lim sup E₁) ≥…

A: Given that if X, Μ, μ  is a measure space and Ej1∞⊂M .

Q: We know that the set S = {1/n : n ∈ N} is not compact because 0 is a limit point of S that is not in…

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Q: Prove that lim n = p.

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Q: If f is continuous and {an} is a sequence such that the limit lim an = L n➔ oo exists, then lim f…

A: Given: f is continuous at L.  an is an sequence such that limn→∞ an=L exists Now we have to prove…

Q: If (an) has limit −1 and (bn) tends to infinity, does it follow that only finitely many terms of…

A: Let an=1,    n=1,2,3−1,  n=4,5,6,7,.... So, liman=−1 Let bn=n So, limbn=∞ Then, anbn=n,…

Q: We denote by 0 the set of all sequences u = (ux)k=1,2, = (u1, u2, ...) (uk E R) %3D satisfying sup…

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Q: Let A be a nonempty subset of R that is bounded above and let α = sup A. If α is NOT an element in…

A: Let A be a nonempty subset of R that is bounded above and let α = sup A. If α is NOT an element in…

Q: Let S be a bounded below infinite subset of R. If w= inf(S), then there exists a monotone sequence…

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Q: Prove that for any bounded sequences (xn) and yn we have lim sup(xn + Yn) < lim sup(xn) + lim…

A: We have to prove that limsupxn+yn≤limsupxn+limsupyn. Let xn and yn are sequence of R. Let limsupxn=x…

Q: Theorem 2.31. In the standard topology on R", if p is a limit point of a set A, then there is a…

A: Considering p as a limit point of set A in the standard topology on  ℝn. Here Kelley's Lemmas…

Q: (c) lim (1+2x)/(2 In z)

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Q: (a) Show that if x is an accumulation point of AC M, then there is a sequence of points of A that…

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Q: Let S be a subset of the metric space M = (X, d), with point p E X. Question 4. Prove that p is a…

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Q: Determine if each of the following statements is true(T) or false(F). 7- In R with the usual metric,…

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Q: Xn+e x1 = 1; xn+1 8

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Q: Prove that a subset of a metric space is closed if and only if contains all its cluster points. Let…

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Q: 8. If A is not totally bounded, show that A has an infinite subset B that is homeo- morphic to a…

A: 8. It is given that A is not totally bounded.

Q: Let X be a set, and define d(x,y) ={0, if x=y; 1, otherwise (a) Show that the function d defines a…

A: Note:  According to bartleby we have to answer only first question please upload the question…

Q: If ∞ is a cluster point of S ⊂ R if for every M ∈ R, there exists an x ∈ S such that x ≥ M.…

A: Given: If ∞ is a cluster point of S⊂R, then for every M∈R, there exists an x∈S such that…

Q: Determine the following limit: In(12æ) lim %3D (12x

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Q: Let an = n n + 1 . Find a number M such that: (a) |an − 1| ≤ 0.001 for n ≥ M. (b) |an − 1| ≤ 0.00001…

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Q: 8. If (X, M, ) is a measure space and {E₁} CM, then (lim inf E₁) ≤ lim inf μ(E). Also, μ(lim sup E₁)…

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Q: 3 Prove the limit At en nti exists does not or exist lim ay tisinas lim a) b)

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Q: In formally proving that lim. (x² + x) = 42, let ɛ > 0 be arbitrary. Choose 8 min m Determine m. m =

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