s5 - Math 5615H: Introduction to Analysis I. Homework #5....

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Math 5615H: Introduction to Analysis I. Fall 2011 Homework #5. Problems and Solutions. #1. Show that for an arbitrary sequence E 1 ,E 2 ,... of sets, the set liminf n →∞ E n := [ n =1 \ k = n E k · is contained in lim sup n →∞ E n := \ n =1 [ k = n E k · . Proof. For arbitrary natural numbers n 1 and n 2 , take n := max { n 1 ,n 2 } . Then \ k = n 1 E k \ k = n E k [ k = n E k [ k = n 2 E k , and [ n 1 =1 \ k = n 1 E k · \ n 2 =1 [ k = n 2 E k · . Remark. x liminf E n ⇐⇒ x E n for all n starting from some n 0 = n 0 ( x ). x limsup E n ⇐⇒ x E n for infinitely many n . #2. Let E be a nonempty open subset of a metric space ( X,d ), which is dense in X . Is it true in general that E = X ? Either prove that E = X , or give a counterexample. Solution. A counterexample: E := (0 , 1) X := [0 , 1] with distance d ( x,y ) := | x - y | . #3.
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s5 - Math 5615H: Introduction to Analysis I. Homework #5....

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