hw1-solns

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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online