JhalyssaWilliams.Analysis.HW2 - Section 3.3 2 a True Every non-empty set of has a min Let S be a non-empty set of If n S then S has a least element

JhalyssaWilliams.Analysis.HW2 - Section 3.3 2 a True Every...

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Section 3.3 2) a. True, Every non-empty set of has a min. Let S be a non-empty set of . If n S, then S has a least element. Suppose n for every m , when m n-1. If m S, then there is a least element in S. b. False; Archimedean Property of states that if were bounded above then there must exist a sup , called a such that a . a is the least upper bound of . Suppose there exist an n such that n> a-1, where a-1 is not the sup . Then n+1 > a , and since n+1 . This contradicts the statement that a is the sup . Therefore a cannot be the sup . c. False, if x=y then xy=x*x=x 2 . Take x= a . Then xy= a 2 ¿ =a, which is rational. d. True, Suppose that x>0. Then there exist an n such that n > 1/(y-x) ny-nx>1 Ny> 1+nx Nx+1<ny When nx > 0. e. False f. False 3) a. Sup=Max =3 b. Sup = Max = π c. Sup = Max = 4 d. Sup=4; No max e. Sup=Max=1/2 f. Sup = Max = 0 g. Sup=1; No max h. Sup= max= 3/2 i. No Sup; No max j. Sup= 4; No max k. Sup=max=1
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l. Sup = 2; No Max m. Sup = 5; No Max n. Sup =
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  • Fall '08
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  • Empty set, Supremum, Order theory, sup, infimum

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