Mark all the true statements about sequences of real numbers. If the limit superior of a bounded sequence equals the limit inferior, then the sequence converges. Every unbounded sequence has a divergent subsequence. If a sequence has no bounded subsequence, then the sequence diverges. If a sequence has no divergent subsequence, then the sequence converges. If a sequence is bounded above, then there is a subsequence that converges to the limit superior of the sequence.

Q: 6. Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence…

A: We have to examine the given sequence abouts its property

Q: Which of the following statements is false? There are no divergent bounded sequences. There are no…

A:

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: Which one of the following statements is CORRECT? O A. Every bounded above monotone sequence…

A: A is correct. For false I have given counter example

Q: 1 Let a1 = 1, and an+1 = 3 An Determine whether the sequence {an} converges or diverges. If it…

A:

Q: Suppose an is a sequence which converges to a real number L. Suppose also that K is a real number…

A: Given that sequence (an) converges to real number L.a If an ⩾ K for all n∈ ℕ then L, whether L⩾Kb If…

Q: i. Every bounded sequence of real numbers converges. ii. Every convergent sequence of real numbers…

A:

Q: Prove that each convergent sequence of reals is bounded.

A:

Q: 14. Prove that every Cauchy sequence is bounded (Theorem 1.4)

A: To prove: Every Cauchy sequence is bounded.

Q: Sequences, limits, Cauchy criterion. a) State the Cauchy condition: A sequence (x,) is a Cauchy…

A: Since the questions contains many subparts as per the guidelines I am answering the first three…

Q: Let (2) be a bounded sequence. Prove that if (en) is monotonically decreasing then it con- verges to…

A: To prove that a bounded, monotonically decreasing sequence {xn​} converges to its infimum, we need…

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

A: We have given few statement . a) If xn is unbounded, then xn has no limit. b) If xn is not monotone,…

Q: Given a sequence (xn) and a real number L , prove or give a counterexample for the following…

A: It is given that:

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

A:

Q: Every increasing monotonic and sequence is convergent

A: given the sequence is increasing monotonic

Q: Consider the sequence {10, -2, ) Find a formula for the general term an of the sequence, assuming…

A:

Q: Prove that there is no sequence of real numbers whose set of accumulation points is exactly Q

A: Answer given below.

Q: Let (an) be an increasing sequence of real numbers bounded from above and (bn) be a decreasing…

A: Given an is an increasing sequence of real numbers bounded above and bn  is a decreasing sequence.…

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…

A: Let (an) be a sequence of positive real numbers.  Case 1: Let (an) is bounded above.  Since (an) be…

Q: Which of the following statements is true? O A monotonic convergent sequence is bounded. O A bounded…

A: Some statements are given, we have to find the true statement

Q: Which of the following statements are false? I. Every convergent sequence is bounded and monotonic.…

A: solution

Q: According to the Bolzano-Weierstrass theorem, every bounded sequence of real numbers has a…

A: Solution :- True Explanation :-

Q: Exercise 2. Let {an}-1 be a non-decreasing (resp. non-increasing) sequence which con- verges to a.…

A: Consider the sequence an be a  non decreasing sequence which converges to a. The sequence sequence…

Q: Which of the following statements is true? O A monotonic convergent sequence is bounded. O A bounded…

A: A real sequence is said to be monotonic if it is either increasing or decreasing But not both.