hmw6s - 18.100B Fall 2002 Homework 6 Due by Noon Tuesday...

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18.100B, Fall 2002, Homework 6 Due by Noon, Tuesday October 29. Rudin: (1) Chapter 4, Problem 20 If E is a nonempty subset of a metric space X, define the distance from x X to E by ρ E ( x )= inf d ( x, z ) . z E ¯ (a) Prove that ρ E ( x ) = 0 if and only if x E. (b) Prove that ρ E is uniformly continuous on X by showing that | ρ E ( x ) ρ E ( y ) |≤ d ( x, y ) for all x, y X. Solution. (a) If ρ E ( x ) = 0 then there exists a sequence z n E such that ¯ d ( x, z n ) 0 . This implies z n x and hence x E. Conversely if ¯ x E then either x E, in which case ρ E ( x )=0 , or else x E ± , so there exists a sequence z n E with z n x. This implies d ( x, z n ) 0 so ρ E ( x )=0 . (b) If x, y X then for any z E, using the triangle inequality ρ E ( x ) d ( x, z ) d ( x, y )+ d ( y, z ) . Taking the infimum over z E on the right-hand side shows that ρ E ( x ) ρ E ( y ) d ( x, y ) . Interchanging the roles of x and y gives the desired estimate | ρ E ( x ) ρ E ( y ) |≤ d ( x, y ) . This proves the uniform continuity of ρ E , since given ±> 0 ,d ( x, y ) implies | ρ E ( x ) ρ E ( y ) | <±. (2) Chapter 4, Problem 23 Area lva luedfunct iondefinedon( a, b ) is said to be convex if f ( λx +(1 λ ) y ) λf ( x )+(1 λ ) f ( y ) whenever x, y
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This note was uploaded on 12/16/2011 for the course STAT 5446 taught by Professor Frade during the Fall '09 term at FSU.

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hmw6s - 18.100B Fall 2002 Homework 6 Due by Noon Tuesday...

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