Unformatted text preview: a n ,b n ). #3. (14 points). Let X be a metric space of bounded sequences x = { x 1 ,x 2 ,...,x n ,... } with the distance d ( x,y ) := sup n  x ny n  for x = { x n } , y = { y n } . Show that the set K := { x = { x n } ∈ X :  x n  ≤ c n = const for all n = 1 , 2 ,... } is compact in X if an only if c n → 0 as n → ∞ . #4. (12 points). If a 1 = 1, and a n +1 = 1 1 + a n for n = 1 , 2 ,..., prove that the sequence { a n } converges and ﬁnd its limit. You can use without proof the fact that lim n →∞ q n = 0 if  q  < 1 ....
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 Fall '09
 Math, Topology, Metric space, disjoint subsets

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