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: 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…

Q: Consider the sequence a. Write out the first five terms of the sequence. a1 = a2 = a3 = a4= a5 = 9.…

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

A: Let Zn be a sequence of complex numbers that converges to C.  Then for ε=1&gt;0, there exists an N∈N…

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

A:

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

A:

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.