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MIT18_014F10_pset1

# MIT18_014F10_pset1 - integer n suppose n< 2 m for some...

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Unit 1: Real Numbers Pset1 Due September 17 (4 points each) 1. Prove Theorem I.11: If ab = 0 then a = 0 or b = 0 2. Prove Theorem I.25: If a < c and b < d then a + b < c + d . 3. Apostol page 43: 1j 4. Course Notes: A Prove Theorem 6 5. Course Notes: A Prove Theorem 12 6. Course Notes: A.10:6 Bonus: (Only to be attempted once other problems are completed) Let A n = x 1 + x 2 + . . . + x n , G n = ( x 1 x 2 . . . x n ) 1 /n n represent the arithmetic and geometric mean, respectively, for a set of n positive real numbers. Prove that G n A n for n = 2. Use induction to show G n A n for any n = 2 k where k is a positive integer. Now for any positive

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Unformatted text preview: integer n , suppose n < 2 m for some integer m . Using the set { x 1 ,x 2 ,...,x n ,A n ,A n ,...,A n } where the A n appears 2 m − n times in the set, show that G n ≤ A n . 1 MIT OpenCourseWare http://ocw.mit.edu 18.014 Calculus with Theory Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms ....
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