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Unformatted text preview: ECEN 601: Assignment 3 Problems: 1. (TOP: 2.7.2) Suppose that f : X Y is continuous. If x is a limit point of the subset A of X , is it necessarily true that f ( x ) is a limit point of f ( A )? 2. (TOP: 2.9.8) Show that the Euclidean distance d on R n is a metric, as follows: If x , y R n and c R , define x + y = ( x 1 + y 1 ,...,x n + y n ) c x = ( cx 1 ,...,cx n ) , x y = x 1 y 1 + + x n y n bardbl x bardbl = ( x x ) 1 / 2 . (a) Show that x ( y + z ) = ( x y ) + ( x z ). (b) Show that | x y | bardbl x bardblbardbl y bardbl . [Hint: If x , y negationslash = 0, let a = 1 / bardbl x bardbl and b = 1 / bardbl y bardbl , and use the fact that bardbl a x b y bardbl 0.] (c) Show that bardbl x + y bardbl bardbl x bardbl + bardbl y bardbl . [Hint: Compute ( x + y ) ( x + y ) and apply previous result.] (d) Verify that the Euclidean distance d ( x , y ) is a metric....
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This note was uploaded on 09/26/2011 for the course ECEN 601 taught by Professor Staff during the Fall '08 term at Texas A&M.
- Fall '08