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Unformatted text preview: k ± k x k − 1 y n − k = n ( x + y ) n − 1 . Now plugin x = y = 1 to get n ² k =0 ° n k ± k 1 k − 1 1 n − k = n (1 + 1) n − 1 = n · 2 n − 1 , which is precisely the statement (since 1 k − 1 1 n − k = 1 for all k and n ). ° 17. OCTOBER 27TH: THE BINOMIAL THEOREM. DEDEKIND CUTS, I. 43 17.3 . Prove that the function Q → D deFned in § 17.5 is injective: that is, prove that if q 1 ° = q 2 , then { x ∈ Q  x < q 1 } ° = { x ∈ Q  x < q 2 } . Answer. Without loss of generality we may assume q 1 < q 2 . Then q 1 °∈ { x ∈ Q  x < q 1 } (since q 1 ° < q 1 ), but q 1 ∈ { x ∈ Q  x < q 2 } (since q 1 < q 2 ). Thus the sets di±er, since they do not have the same elements. °...
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This note was uploaded on 12/14/2011 for the course MGF 3301 taught by Professor Aluffi during the Fall '11 term at FSU.
 Fall '11
 Aluffi

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