exercises_6 - Analysis 1 Exercises 6 1 Prove that ∀ a>...

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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 Cauchy-Bunyakovski-Schwarz 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.

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exercises_6 - Analysis 1 Exercises 6 1 Prove that ∀ a>...

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