For any sequence (xn) of real numbers, we say that (xn) is increasing iff for all...

For any sequence (x_n) of real numbers, we say that (x_n) is increasing iff for all n, m ∈ N, if n < m, then x_n < x_m. Prove that any increasing sequence that is not Cauchy must be unbounded. (Here, “unbounded” just means that x_n eventually gets larger than any given real number). Then, show that any increasing sequence that is bounded must converge

Expert Solution

Colby Messinger answered 7 months ago

A sequence is just an infinite list of numbers (say real numbers, we often denote these...

A sequence is just an infinite list of numbers (say real numbers, we often denote these by a0,a1,a2,a3,a4,.....,ak,..... so that ak denotes the k-th term in the sequence. It is not hard to see that the set of all sequences, which we will call S, is a vector space. a) Consider the subset, F, of all sequences, S, which satisfy: ∀k ≥ 2,a(sub)k = a(sub)k−1 + a(sub)k−2. Prove that F is a vector subspace of S. b) Prove that if...

We say that an infinite sequence a0,a1,a2,a3,… of real numbers has the limit L if for...

We say that an infinite sequence a0,a1,a2,a3,… of real numbers has the limit L if for every strictly positive number ε, there is a natural number n such that all the elements an,an+1,an+2,… are within distance ε of the value L. In this case, we write lim a = L. Express the condition that lim a = L as a formula of predicate logic. Your formula may use typical mathematical functions like + and absolute value and mathematical relations like...

Let {xn} be a real summable sequence with xn ≥ 0 eventually. Prove that √(Xn*Xn+1) is...

Let {xn} be a real summable sequence with xn ≥ 0 eventually. Prove that √(Xn*Xn+1) is summable.

Let X = {x1,x2,...,xn} a sequence of real numbers. Design an algorithm that in linear time...

Let X = {x1,x2,...,xn} a sequence of real numbers. Design an algorithm that in linear time finds the continue subsequence of elements xi,xi+1,...,x, which product is the maximum. Suppose that the product of an empty subsequence is 1 and observe that the values can be less to 0 and less to 1.

Given an array A of n distinct real numbers, we say that a pair of numbers...

Given an array A of n distinct real numbers, we say that a pair of numbers i, j ∈ {0, . . . , n−1} form an inversion of A if i < j and A[i] > A[j]. Let inv(A) = {(i, j) | i < j and A[i] > A[j]}. Answer the following: (a) How small can the number of inversions be? Give an example of an array of length n with the smallest possible number of inversions. (b)...

a real sequence xn is defined inductively by x1 =1 and xn+1 = sqrt(xn +6) for...

a real sequence xn is defined inductively by x1 =1 and xn+1 = sqrt(xn +6) for every n belongs to N a) prove by induction that xn is increasing and xn <3 for every n belongs to N b) deduce that xn converges and find its limit

Let A be a set of real numbers. We say that A is an open set...

Let A be a set of real numbers. We say that A is an open set if for every x0 ∈ A there is some δ > 0 (which might depend on x0) such that (x0 − δ, x0 + δ) ⊆ A. Show that a set B of real numbers is closed if and only if B is the complement of some open set A

4.2.1. Problem. Suppose that (xn) and (yn) are sequences of real numbers, that xn → a...

4.2.1. Problem. Suppose that (xn) and (yn) are sequences of real numbers, that xn → a and yn → b, and that c ∈ R. For the case where a and b are real numbers derive the following: (a) xn +yn →a+b, (b) xn −yn →a−b, (c) xnyn → ab, (d) cxn → ca, (e) xn/yn → a/b if b ̸= 0.

. Suppose that the sequence (xn) satisfies |xn –α| ≤ c | xn-1- α|2 for all...

. Suppose that the sequence (xn) satisfies |xn –α| ≤ c | xn-1- α|2 for all n. Show by induction that c | xn- α| ≤ c | x0 - α|2n , and give some condition That is sufficient for the convergence of (xn) to α. Use part a) to estimate the number of iterations needed to reach accuracy |xn –α| < 10-12 in case c = 10 and |x0 –α |= 0.09.

Consider the sequence: x0=1/6 and xn+1 = 2xn- 3xn2 | for all natural numbers n. Show:...

Consider the sequence: x0=1/6 and xn+1 = 2xn- 3xn2 | for all natural numbers n. Show: a) xn< 1/3 for all n. b) xn>0 for all n. Hint. Use induction. c) show xn isincreasing. d) calculate the limit.

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