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Unformatted text preview: Analysis 1: Exercises 6 1. Prove that ( ∀ a > 0) a + 1 a ≥ 2 . 2. Prove that ( ∀ a,b,c ≥ 0) ( a + b )( a + c )( b + c ) ≥ 8 abc . 3. Prove that ( ∀ n ∈ N ) ( n ≥ 2) ⇒ 1 2 2 + 1 3 2 + ··· + 1 n 2 < 1 . 4. Let x,y ∈ R . Show that a)   x  =  x  , x 6  x  ,  x  = max { x, x } ; b)  xy  =  x  y  ; c)  x + y  6  x  +  y  ; d)  x    y  6  x y  . 5. Prove that ( ∀ n ∈ N + )( ∀ a 1 ,a 2 ,...,a n ∈ R )[  a 1 + a 2 + ··· + a n  ≤  a 1  +  a 2  + ··· +  a n  ] . 6. Prove the CauchyBunyakovskiSchwarz inequality: ( ∀ n ∈ N + )( ∀ a 1 ,a 2 ,...,a n ∈ R ) ( ∀ b 1 ,b 2 ,...,b n ∈ R ) n X k =1 a k b k 2 ≤ n X k =1 a 2 k n X k =1 b 2 k . Hint : For all x ∈ R , n X k =1 ( a k x + b k ) 2 = x 2 n X k =1 a 2 k + 2 x n X k =1 a k b k + n X k =1 b 2 k ≥ . 7. Let a 1 ,a 2 ,...,a n be positive real numbers. Their arithmetic mean A n and harmonic mean H n are defined by A n = a 1 + a 2 + ... + a n n , H 1 n = 1 n 1...
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This note was uploaded on 02/25/2010 for the course MATHEMATIC 11007 taught by Professor Spyros during the Winter '10 term at Bristol Community College.
 Winter '10
 Spyros

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