hw1-solns - Math 104 - Homework 1 Solutions Lectures 2 and...

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Math 104 - Homework 1 Solutions Lectures 2 and 4, Fall 2011 1. Ross, 3.5. (a) Show that | b | ≤ a if and only if - a b a . (b) Prove that || a | - | b || ≤ | a - b | for all a,b R . Proof. (a) Suppose that | b | ≤ a . Note that this implicitly includes the assumption that a 0. There are two cases to consider. First, if b 0, then | b | = b so our assumptions is precisely that b a . Since - a 0 b , we have - a b a as required. Instead, if b < 0, then | b | = - b , so our assumptions becomes - b a, or b ≥ - a. Since b < 0 a , we again have - a b a . Thus in either case - a b a . Conversely suppose that - a b a . If b 0, then | b | = b and b a gives us | b | ≤ a . If b < 0, then | b | = - b and - a b gives a ≥ - b = | b | . Thus again in either case we have | b | ≤ a . (b) By part (a), to establish the required inequality it is enough to show that -| a - b | ≤ | a | - | b | ≤ | a - b | . (In other words, apply part (a) with | a | - | b | playing the role of “ b ” and | a - b | playing the role of a ”.) From the triangle inequality, we know that | a | = | ( a - b ) + b | ≤ | a - b | + | b | , so | a | - | b | ≤ | a - b | . Switching the roles of a and b shows that also | b | - | a | ≤ | b - a | ; since | b - a | = | a - b | , this gives | a | - | b | ≥ -| a - b | . Thus -| a - b | ≤ | a | - | b | ≤ | a - b | as required. 2. Ross, 3.6. (a) Prove that | a + b + c | ≤ | a | + | b | + | c | for all a,b,c R . (b) Use induction to prove | a 1 + a 2 + ··· + a n | ≤ | a 1 | + | a 2 | + ··· + | a n | for n numbers a 1 ,a 2 ,...,a n . Proof. (a) By the triangle inequality, we have | a + b + c | = | a + ( b + c ) | ≤ | a | + | b + c | . Again by the triangle inequality, | b + c | ≤ | b | + | c | , so | a + b + c | ≤ | a | + | b + c | ≤ | a | + | b | + | c | as claimed. (b) The base case n = 2 is the statement of the triangle inequality. By way of induction, suppose that the statement is true for some k : i.e. suppose that | a 1 + ··· + a k | ≤ | a 1 | + ··· + | a k | . Then
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hw1-solns - Math 104 - Homework 1 Solutions Lectures 2 and...

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