Question: Let S be a set that is bounded below. Prove that a lower bound w of S is the infimum of S if and only

Let S be a set that is bounded below. Prove that a lower bound w of S is the infimum of S if and only if for any ε > 0 there exists t ∈ S such that t < w + ε.

Step by Step Solution

★★★★★

3.32 Rating (161 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

If w inf S and 0 then w is not a lower bound so that ... 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)

829-C-I (949).docx

120 KBs Word File

Students Have Also Explored These Related Calculus Questions!

Let S be a set of five positive integers the maximum of which is at most 9. Prove that the sums of the elements in all the nonempty subsets of S cannot all be distinct.

Let S be a set of seven positive integers the maximum of which is at most 24. Prove that the sums of the elements in all the nonempty subsets of S cannot be distinct.

Let S be a set containing n distinct objects. Verify that ex/(I - x)k is the exponential generating function for the number of ways to choose m of the objects in S, for 0

Show work/explanation where indicated. Answers without any work may earn little, if any, credit. You may type or write your work in your copy of the exam, or if you prefer, create a document...

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...

Could someone please check my work 1. If S1 , and S2 , are compact subsets of R , prove that S1 U S2 is compact. Suppose S1 and $2 are compact subsets of R . By Theorem 3.5.5 (Heine Borel), since S1...

6. (a) Prove that a lower bound of a set need not be unique but the infimum of a given set E is unique. (b) Prove that if E is a nonempty subset of R that is bounded below, then E has a finite...

Prove that a lower bound of a set need not be unique but the infimum of a given set E is unique.

10. (16%) For each of the following statements, determine whether it is true or false. Then give a brief justification for your answer. Your answer will be evaluated based on your justification and...

Repeat Exercises 10.4 through 10.6 using the unequal variances procedure. Compare your results to those obtained using the equal variances procedure. In Exercises 10.4 Calculate a 95 percent...

Decomposition of X exhibits a rate constant of 0.05 mg/year. How many years are required for the decomposition of 5 mg of X into 2.5 mg (a) 25 (b) 50 (c) 20 (d) 40

Riskier firms have a higher risk - adjusted cost of capital which means that the discount rate for capitalizing their earnings will be Multiple choice question. lower, resulting in lower share...

Using an online networking platformfor example, LinkedInnetwork and connect with someone in product development. Explain that you have a college project and that youd like to interview them. There...

Find the volume of the solid obtained by rotating about the x-axis the region enclosed by the curves y = 9/(x2 + 9). y = 0, x = 0, and x = 3

(a) Use trigonometric substitution to verify that (b) Use the figure to give trigonometric interpretations of both terms on the right side of the equation in part (a). Va? - dt =a sin (x/a) + x/a? x2

A torus is generated by rotating the circle x2 + (y - R)2 = r2 about the x-axis. Find the volume enclosed by the torus.

Professors Annuity Corp. offers a lifetime annuity to retiring professors. For a payment of $78,000 at age 65, the firm will pay the retiring professor $550 a month until death. a. If the professors...

Your client, Tom, has come to you inquiring about retirement. He wants to know approximately how much (in future dollars) he will need to have saved by retirement. You have calculated that he will...

Q1. What is the present value of the cost of college education for 4 children ages 1,3,5, and 7. The current cost of college is $25,000. The children will begin college at age 18 and be in college...