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# hw 1 - an extreme point Exercise 4 Show that if x n ∞ n...

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Department of Mathematical Sciences University of Delaware Prof. T. Angell September 15, 2010 Mathematics 530 Exercise Sheet 1 Exercise 1 : Show that, if C R n is a convex set then, for any x, y, z C and μ i 0 with 3 i =1 μ i = 1, it is true that μ 1 x + μ 2 y + μ 3 z C . (DON’T just quote a theorem, check this result “from scratch”.) Exercise 2 : Let A and B be nonempty convex subsets of R n . Show that their Cartesian product A × B is also convex in R 2 n . Exercise 3 : Sketch the region P in R 2 determined by the set of inequalities x 1 + x 2 4 2 x 1 + x 2 6 x 2 3 x 1 , x 2 0 Show that this set is convex and identify all those points x o P which have the property that there are no points x 1 , x 2 P with x 1 6 = x 2 so that x o can be written as x o = (1 - λ ) x 1 + λ x 2 . Such points are called extreme points of the set P . They are important,
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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 2-y 2 = ( x + y )( x-y ) 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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