Unformatted text preview: an extreme point. Exercise 4 : Show that if { x n } ∞ n =1 is a sequence of real numbers that converges to the point x o ∈ R then  x n  →  x o  as n → ∞ . Otherwise said, the absolute value function is continuous. Exercise 5 : Show that the function f ( x ) = x 2 is continuous at any real x o by showing that if { x n } ∞ n =1 is any sequence with lim n →∞ x n = x o then lim n →∞ f ( x n ) = f ( x o ). (HINT: x 2y 2 = ( x + y )( xy ) and, since x n → x o , then for n suﬃciently large,  x n + x o  is bounded by 3  x o  .)...
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 Fall '08
 Luke,D
 Math, Continuous function, Prof. T. Angell, Department of Mathematical Sciences University of Delaware

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