A sequence (xn)∞ {n=1} is said to oscillate if lim inf xn < lim sup xn.  Prove or disprove the following.  a) Every oscillating sequence diverges. b) Every divergent sequence oscillates. c) Every oscillating sequence has a convergent subsequence.

Q: Prove that limx → c f(x) = ∞ if and only if for every sequence an that is in the domain of f and…

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Q: Prove that limx → c f(x) = ∞ if and only if for every sequence an that is in the domain of f and…

A: As per the question we have to prove that limx → c f(x) = ∞ if and only if for every sequence an…

Q: Give a counterexample and explain why this statement is false: “If a sequence contains both a…

A: The given statement statement “If a sequence contains both a strictly increasing subsequence and a…

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A: here we have to check whether the sequence is convergent or not.

Q: 00 A student claims that converges because n= lim -15

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Q: B. Prove that limx → c f(x) = o if and only if for every sequence aŋ that is in the domain of f and…

A: Consider a function fx. Suppose that limx→cf(x)=∞. That is for all N&gt;0, there exists a δ&gt;0,…

Q: If {a} be a sequence such that lim an +1 an = 1, where | 1|< 1, then lim an = 0.

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Q: 3. Determine if the sequence xn = 47² is convergent or not and find its limit if it exists n²+1

A: We want to find the given sequence is convergent or divergent.

Q: 1 3 .5 4 X +x

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Q: Show that any convergent sequence of complex numbers is bounded.

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Q: Let (an) be a sequence of real numbers that diverges to +∞, and let (bn) be bounded. Prove that lim…

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Q: } be a sequence of real numbers. } is unbounded, then {xn} has no limit. 2} is not monotone, then…

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Q: C. Prove that limx - c f(x) = ∞ if and only if for every sequence an that is in the domain off and…

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Q: 6n Determine if the sequence an 4(-1) converges or diverges. If it converges, compute its limit. If…

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Q: lim 3n+2 2n + 1 = 3/2

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Q: Let a < b be real numbers and assume a sequence (Tk)ž is strictly decreasing and Xk E (a, b) for…

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Q: Give the definition of a subsequence of a sequence Xn.

A: the definition of a subsequence of a sequence Xn.

Q: If lim(x,) =5100, then (Xn) is bounded. Select one: O True O False

A: 1) Here, lim (xn)=5100 that is finite. That is the sequence xn is bounded. So, the statement is…

Q: Solve all 1) Assume that (xn) is a monotone decreasing and bounded sequence. Prove that (xn) is…

A: is monotone decreasing and bounded sequence.We have to prove that  is convergent.

Q: Show that any sequence of positive real numbers either has a subsequence that converges, or else a…

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Q: Show that any sequence of positive real numbers either has a subsequence that converges, or else a…

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Q: (-1)" :n Determine if A is bounded above, bounded 2. Let A = n below, or both. If it is bounded…

A: Bounded above. Bounded below.