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# p3 - 2#4 Let x any y be vectors in R k Show that y is...

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Math 5615H: Introduction to Analysis I. Fall 2011 Homework #3 (due on Wednesday, September 28). 50 points are divided between 5 problems, 10 points each. #1. Let A := { a 1 ,a 2 ,... } be a set of real numbers deﬁned as follows: a 1 = 1 , and a k +1 = 1 + a k for k = 1 , 2 ,.... Find sup A . #2. Let z 1 and z 2 be complex imaginary (not real) numbers such that both z 1 + z 2 and z 1 z 2 are real . Show that z 2 = z 1 , i.e. z 2 is the conjugate of z 1 . #3 (exercise 14 on p.23). If z is a complex number such that | z | = 1, compute | 1 + z | 2 + | 1 - z |
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Unformatted text preview: 2 . #4. Let x any y be vectors in R k . Show that y is uniquely represented in the form y = a + b, where a,b ∈ R k satisfy a = αx for some real α, and b · x = 0 . Verify this fact for x = (1 , 1 , 1) and y = (1 , 2 , 3) in R 3 . #5. Let A be a nonempty set in R k . Deﬁne d ( x ) := inf {| x-a | : a ∈ A } . Show that | d ( x )-d ( y ) | ≤ | x-y | for all x,y ∈ R k ....
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