hw11solutions - Math 521 - Advanced Calculus I Instructor:...

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Math 521 - Advanced Calculus I Instructor: J. Metcalfe Due: February 17, 2010 Assignment 11 1. Let E 0 be the set of all limit points of a set E . Prove that E 0 is closed. Do E and E 0 always have the same limit points? For the latter question, the answer is no. Look at E = { 1 /n : n N } . Here E 0 = { 0 } , but there are no limit points of E 0 . To show that E 0 is closed, we need to show that it contains its limit points. Thus, we let x be a limit point of E 0 and we need to show that x is a limit point of E (i.e. is in E 0 ). For a given ε > 0, we need to show that there is a point y E with y 6 = x so that y V ε ( x ). For this same ε , since x is a limit point of E 0 , we know that there is an element z E 0 with z 6 = x so that z V ε/ 2 ( x ). I.e. | z - x | < ε/ 2. Since z E 0 (i.e. z is a limit point of E ), there is an element y E with y 6 = z so that y V | x - z | / 2 ( z ). Then, | x - y | ≤ | x - z | + | z - y | < ( ε/ 2) + | x - z | / 2 < 3 ε/ 4 < ε. Thus,
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