# ps4 - = x Part 2 3(a Show that if a n n ∈ N is a...

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18.100B and 18.100C Fall 2011 Problem Set 4 Due October 13th at 4 pm in room 2-108. Hand in parts 1, 2 and 3 separately. Put your name and whether you are registered for 18.100B or 18.100C on each part. Part 1 1. Problem 1 from page 78. 2. Let X be a complete metric space with metric d , and let f : X X be a contraction , meaning that there exists λ < 1 such that d ( f ( x ) ,f ( y )) λd ( x,y ) for all x,y X . Prove that there is a unique point x 0 X such that f ( x
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Unformatted text preview: ) = x . Part 2 3. (a) Show that if ( a n ) n ∈ N is a convergent sequence of nonnegative real numbers then lim n →∞ √ a n = q lim n →∞ a n (b) Problem 2 from page 78 Part 3 4. Let K be a compact metric space, and { G α } α ∈ A an open cover of K . Prove that there exists ε > 0 such that for every x ∈ K there exists α ∈ A such that N ε ( x ) ⊂ G α . 1...
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