Question: Suppose that E R is a nonempty bounded set and that sup E E. Prove that there exists a strictly increasing sequence {xn}

Suppose that E ⊂ R is a nonempty bounded set and that sup E ∉ E. Prove that there exists a strictly increasing sequence {xn} that converges to sup E such that xn ∊ E for all n ∊ N.

Step by Step Solution

★★★★★

3.36 Rating (183 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

Choose by the Approximation Property a n x 1 E such that supE 1 x 1 supE Since s... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Document Format (1 attachment)

741-M-N-A-D-I (194).docx

120 KBs Word File

Students Have Also Explored These Related Numerical Analysis Questions!

If S R is a nonempty bounded set, and IS := (inf S; sup S) , show that S IS. Moreover, if J is any closed bounded interval containing S, show that IS J.

Decide which of the following statements are true and which are false. Prove the true ones and give counterexamples to the false ones. a) If A and B are nonempty, bounded subsets of R, then sup(A B)...

Show that if E is a nonempty bounded subset of Z, then inf E exists and belongs to E.

2. Suppose that E c R is a nonempty bounded set and sup E ~ E. Prove that there exists a strictly increasing sequence {xn} that converges to sup E such that Xn E E for all n E N.

Recall that N is the set of positive integers and Q is the set of rational numbers. NOTE: "Limit point\Sequences - Part 2 (Week 4) Infinite Limits Definition. A sequence (sn) is said to diverge to +...

2. (a) Let B be a non-empty subset of real numbers that is bounded below with lower bound B E R. Prove that B = inf(B) if and only if there exists a sequence (bn) such that bn E B for each n E N and...

1. A player throws a fair die and simultaneously flips a fair coin. If the coin lands heads, then she wins twice, and if tails, then one-half of the value that appears on the die. Determine her...

Applied Mathematics and Computation 95 (1998) 181192 Love dynamics: The case of linear couples Sergio Rinaldi 1 Centro Teoria dei Sistemi, CNR, Politecnico di Milano, Via Ponzio 34/5, 20133 Milan,...

Math 125 A: Homework 5 Due Thursday, November 9 1. For n = 1, 2, let fn : E R and suppose that fn f uniformly for some function f . Show that if each fn is bounded, then f is bounded. 2. Let {fn } be...

Write down your proof of the "Monotone Convergence Theorem:" that every bounded monotone(-decreasing) sequence converges (to its infimum). Here's the google doc we worked on together.(Links to an...

Consider the Cobb-Douglas production model for a manufacturing process depending on three inputs x, y, and z with unit costs a, b, and c, respectively, given by P = kx(y(z(, ( > 0, ( > 0, ( > 0, ( +...

In problem, find the indicated derivatives and simplify. 0.4r) (0.5-21)] (3 dt

A security architect is planning a Statement of Work to perform services at various levels of the Risk Management Lifecycle. The security architect should allocate the most hours to which phase?...

(The National Economy)Why do economists pay more attention to national economies (for example, the U.S. or Canadian economies) than to state or provincial economies (such as California or Ontario)?

1.23 Example. Let A = {1,~, i,~, ... } and B = g, ,~, ... }. Prove that sup A = supB = 1.

PROOF. Let SI and S2 be suprema of the same set E. Then both SI and S2 are upper bounds of E, whence by Definition 1. 16iii, SI :::; S2 and S2 :::; SI. We conclude by the Trichotomy Property that s 1...

1.19 Remark. If a set has a supremum, then it has only one supremum.

Arnold Inc. is considering the purchase of a new truck. It can choose between two brands. Brand A: buying the new truck for $120,000. This brand-A truck lasts for (7 + A) years (A: the remainder of...

Lush Gardens Co. bought a new truck for $52,000. It paid $5,200 of this amount as a down payment and financed the balance at 5.44% compounded semi-annually. If the company makes payments of $1,700 at...

roducing three different scenarios. Under a p essimistic scenario the property would produce a B IRRp0f8%; a most-likely scenario produces a BTIRRp of 12%. The investor assigns the pessimistic...