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# presentations1 - 18.100C Presentation topics for 9/10...

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18.100C Presentation topics for 9/10 Problem 1 (Rudin Problems 4,5 pg.22) Review the notion of upper and lower bound, inﬁmum, and supremum, then solve: (i) Let E S be a nonempty subset of an ordered set S . Suppose α S is a lower bound of E and β S is an uppper bound of E . Prove that α β . (ii) Let A R be a nonempty subset of the real numbers. Deﬁne - A = {- x x A } to be the set of all numbers - x , where x A . Show that inf A = - sup( - A ). (Consider separately the cases when A is bounded below and when not.) Problem 2 (Rudin Problem 9 pg.22 – lexicographic order) Review the notion of ordered sets and least-upper-bound property, then solve: For complex numbers z = a + bi C and w = c + di C deﬁne “ z < w ” if either a < c or if ( a = c and b < d ). Prove that this turns C into an ordered set. Is this an ordered ﬁeld? Does it have the least-upper-bound property? Problem 3

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## This note was uploaded on 12/07/2011 for the course MATH 18.100B taught by Professor Prof.katrinwehrheim during the Fall '10 term at MIT.

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presentations1 - 18.100C Presentation topics for 9/10...

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