# HW#2 - SOLUTIONS TO HOMEWORK 2 3.5(a Prove |b| a-a b a...

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SOLUTIONS TO HOMEWORK # 2 3.5 (a) Prove | b | ≤ a ⇔ - a b a . Since | b | ≥ 0 we have a 0. Consider two cases: Case 1: b 0; then | b | = b and we immediately have b ≥ - a . Also | b | ≤ a b a so in this case our statement holds. Case 2: b < 0; then | b | = - b,b a . Also | b | ≤ a ⇔ - b a b ≥ - a so in this case our statement holds too. We are done. (b) Show | a | - | b | a - b . From (a), it will suﬃce to show - a - b ≤ | a | - | b | ≤ a - b . Note | a | = | a + b - b | ≤ | a - b | + | b | using the triangle inequality (1) | b | = | b + a - a | ≤ | b - a | + | a | same reason (2) Thus | a | - | b | ≤ | a - b | , from the top inequality (1), and | b | - | a | ≤ | b - a | = | a - b | , so -| a - b | ≤ | a | - | b | , from (2) . Thus -| a - b | ≤ | a | - | b | ≤ | a - b | , and this proves the claim. 3.7 (a)

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HW#2 - SOLUTIONS TO HOMEWORK 2 3.5(a Prove |b| a-a b a...

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