By Tracy Kompelien

ISBN-10: 159928507X

ISBN-13: 9781599285078

E-book annotation no longer on hand for this title.**Title: **2-D Shapes Are in the back of the Drapes!**Author: **Kompelien, Tracy**Publisher: **Abdo Group**Publication Date: **2006/09/01**Number of Pages: **24**Binding sort: **LIBRARY**Library of Congress: **2006012570

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Also, denote Pn 2 2 2 2 k¼1 bk ¼ b1 þb2 þÁÁÁþbn Pn by B and k¼1 ak bk ¼ a1 b1 þ a2 b2 þÁÁÁþan bn by C. Now, for any real number l, we have that ðlak þ bk Þ2 ! 0, so that n À n X Á X l2 a2k þ 2lak bk þ b2k ¼ ðlak þ bk Þ2 ! 0; k¼1 Note that we will use the Æ notation to keep the argument brief. k¼1 which we may rewrite in the form l2 A þ 2lC þ B ! 0: But this inequality is equivalent to the inequality ðlA þ C Þ2 þ AB ! C2 ; for any real number l: Since A is non-zero, we may now choose l ¼ À CA. It follows from the last & inequality that AB !

Remark A careful examination of the proof of Theorem 3 shows that equality can only occur if all the terms ai are equal. 1 Least upper bounds and greatest lower bounds Upper and lower bounds Any finite set {x1, x2, . , xn} of real numbers obviously has a greatest element and a least element, but this property does not necessarily hold for infinite sets. That is ða1 a2 Þ a3 a4 . . akþ1 ¼ 1 ) ða1 a2 Þ þ a3 þ a4 þ Á Á Á þ akþ1 ! 4 Least upper bounds and greatest lower bounds 23 For example, the interval (0, 2] has greatest element 2, but neither of the sets N ¼ {1, 2, 3, .

Thus it seems likely that sup E is the decimal representation of 2. But how can we prove that (sup E)2 ¼ 2? 5, once we have described how to do arithmetic with real numbers (decimals). Finally, note that there is a corresponding result about lower bounds. The Greatest Lower Bound Property of R Let E be a non-empty subset of R. If E is bounded below, then E has a greatest lower bound. 4 Proof of the Least Upper Bound Property You may omit this proof at a first reading. We know that E is a non-empty set, and we shall assume for simplicity that E contains at least one positive number.

### 2-D Shapes Are Behind the Drapes! by Tracy Kompelien

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