4130sols5 - 4130 HOMEWORK 5 Due Tuesday March 9(1 A subset I of R is called an interval if for all x,y ∈ I and all z ∈ R if x< z< y then z

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 4130 HOMEWORK 5 Due Tuesday March 9 (1) A subset I of R is called an interval if for all x,y ∈ I and all z ∈ R , if x < z < y then z ∈ I . Show that if I is a bounded interval, then (inf I, sup I ) ⊂ I . Using this, show that I must be one of the following four intervals: (inf I, sup I ) [inf I, sup I ) (inf I, sup I ] [inf I, sup I ] . Let I be a bounded interval. Then sup I and inf I exist. Suppose inf I < z < sup I . Then there is some x ∈ I with inf I ≤ x < z . Indeed, if there was no such x , then z would be a lower bound for I , but z is greater than the greatest lower bound inf I . Similarly, there is some y ∈ I with z < y ≤ sup I . Therefore, x < z < y and so z ∈ I by definition of an interval. Therefore, (inf I, sup I ) ⊂ I . Now suppose w < inf I . Then w / ∈ I because inf I is a lower bound for I . Similarly, if w > sup I then w / ∈ I . Therefore, we have I ⊂ [inf I, sup I ]. Altogether, we have (inf I, sup I ) ⊂ I ⊂ [inf I, sup I ] which leaves only the four given possibilities....
View Full Document

This note was uploaded on 09/22/2010 for the course MATH 413 at Cornell University (Engineering School).

Page1 / 3

4130sols5 - 4130 HOMEWORK 5 Due Tuesday March 9(1 A subset I of R is called an interval if for all x,y ∈ I and all z ∈ R if x< z< y then z

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online