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4130sols5 - 4130 HOMEWORK 5 Due Tuesday March 9(1 A subset...

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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.
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