Homework_set_5_typeset

# Homework_set_5_typeset - interval 6,7 8 12 Let a 1 a 2 a 3...

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Math 413, Fall semester, 2007 Bigtimes Homework set 5, handed out September 20, due September 27 Remember that there is a prelim scheduled for October 4, 7:30 pm. I don’t know the room yet, but I will announce it as soon as I do. The curriculum is the text through Chapter 4. Finish reading Chapter 4. In the text, Section 4.1.5, do problems 4,5,7,8, 13 (this is closely related to 7), 15. None of these problems are hard. Section 4.2.4, do problems 3 (I don’t understand the second part of the question; instead, give examples where the domain is an open interval and the range is a closed interval. Do there exist examples where the domain is a closed interval and the image is an open
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Unformatted text preview: interval?), 6,7, 8, 12. Let a 1 , a 2 , a 3 , . . . be an ennumeration of the rationals in [0 , 1]. Choose ² > 0 so that ² < 1 / 8. Consider the union U = [0 , 1] ∩ ∞ [ n =1 ‡ a n-² 2 n , a n + ² 2 n · . a) Show that U is an open in [0 , 1], and dense in [0 , 1]. b) Show that [0 , 1]-U is not empty. Use the speciﬁc list , 1 , 1 / 2 , 1 / 3 , 2 / 3 , 1 / 4 , 3 / 4 , 1 / 5 , 2 / 5 , 3 / 5 , 4 / 5 , . . . , taking all the numbers with denominator 1, then those with denominator 2, then denom-inator 3, etc, alwaqys in increasing order, and omitting any numbers that have already appeared in the list. c) Can you ﬁnd an element of [0 , 1]-U ?...
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