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 n² 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 denominator 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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This homework help was uploaded on 02/23/2008 for the course MATH 4130 taught by Professor Hubbard during the Fall '07 term at Cornell.
 Fall '07
 HUBBARD
 Math

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