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Unformatted text preview: MATH 290 – WRITING ASSIGNMENT 1 – SOLUTIONS 1. Prove that for all real numbers a and b , | a + b | ≤ | a | + | b | . (Hint: This will be an example of a proof by cases.) Solution: Let a and b be real numbers. We must show that | a + b | ≤ | a | + | b | . In order to do this we will first prove the following claim. Claim. For all real numbers x ,-| x | ≤ x ≤ | x | . To prove this claim, let x be a real number. There are two cases. Assume first that x ≥ 0. Since x ≥ 0, | x | = x so that x ≤ | x | . Since x = | x | ≥ 0 we have that- x ≤ 0, hence that- x ≤ ≤ | x | and finally that-| x | ≤ x . Therefore in this case,-| x | ≤ x ≤ | x | . Assume next that x < 0. Since x < 0, | x | =- x , so that-| x | ≤ x . Since | x | ≥ 0 and x < 0, x < ≤ | x | , so that x ≤ | x | . Therefore in this case,-| x | ≤ x ≤ | x | , and the claim is proved. To finish the proof there are two cases to consider. Assume that a + b ≥ 0....
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