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assignment4

# assignment4 - constant K> 0 such that | f x-f y | ≤ K |...

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MATH 255 ASSIGNMENT 4 This assignment is due in class on Monday, February 7 Problems Please justify carefully your answers. 1. [10 points] Let X be the collection of all sequences of positive integers. If x = ( n j ) j =1 and y = ( m j ) j =1 are two elements of X , set k ( x, y ) = inf { j : n j 6 = m j } and d ( x, y ) = ( 0 if x = y 1 k ( x,y ) if x 6 = y. In the Assignment 2 it was shown that d is a metric on X . Prove that the metric space ( X, d ) is complete. 2. [15 points] For x, y R , set d 1 ( x, y ) = | x - y | , d 2 ( x, y ) = | arctan x - arctan y | . It is very easy to show that d 2 is a metric on R (you do not need to provide the proof). (1) Show that ( R , d 1 ) and ( R , d 2 ) have the same convergent sequences (that is, a sequence of real numbers ( x n ) converges in ( R , d 1 ) if and only if it converges in ( R , d 2 )). (2) Show that the sequence x n = n is Cauchy in ( R , d 2 ). (3) Is ( R , d 2 ) complete? 3. [10 points] Which of the following sequence converges in C ([0 , 1]) (w.r.t. the usual sup norm)? If the sequence converges, find its limit. Justify your answers. (1) f n ( x ) = x n . (2) f n ( x ) = x n +1 - x n . 4. [15 points] Recall that a function f : [0 , 1] R is called Lipschitz if there exists a

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Unformatted text preview: constant K > 0 such that | f ( x )-f ( y ) | ≤ K | x-y | for all x,y ∈ [0 , 1]. Denote by Lip([0 , 1]) the collection of all Lipschitz functions on [0 , 1]. It is trivial to show that Lip([0 , 1]) is a vector space with respect to the usual operations 1 of addition and scalar multiplications of functions (you do not need to write the proof). For f ∈ Lip([0 , 1]) set k f k = | f (0) | + sup ≤ x,y ≤ 1 x 6 = y | f ( x )-f ( y ) | | x-y | . (1) Show that k · k is a norm on C ([0 , 1]). (2) Show that (Lip([0 , 1]) , k · k ) is a Banach space. 5. [10 points] Let ( X,d ) be a complete metric space and E n , n = 1 , 2 , ··· , a sequence of closed sets such that diam( E n ) = sup { d ( x,y ) : x,y ∈ E n } < ∞ . If E n ⊃ E n +1 for all n , and if lim n →∞ diam( E n ) = 0, prove that ∩ ∞ n =1 E n consists of exactly one point. 2...
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assignment4 - constant K> 0 such that | f x-f y | ≤ K |...

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