Homework_Set_6_Solution

Homework_Set_6_Solution - Hw 6 solutions 10/31/2005 Problem...

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Unformatted text preview: Hw 6 solutions 10/31/2005 Problem 1a In the following, let I j,n = [ j 2 n , j +1 2 n ] where j { , , 2 n 1 2 n } . J n ( f ) J n +1 ( f ). Note that I 2 j,n +1 I 2 j +1 ,n +1 = I j,n . Hence, sup I j,n ( f ) = max( sup I 2 j,n +1 ( f ) , sup I 2 j +1 ,n +1 ( f )) Thus, 1 2 n sup I j,n ( f ) = 1 2 n +1 sup I j,n ( f ) + 1 2 n +1 sup I j,n ( f ) (1) 1 2 n +1 sup I 2 j,n +1 ( f ) + 1 2 n +1 sup I 2 j +1 ,n +1 ( f ) (2) Summing the inequalities given by(2) over j = 0 , , 2 n 1 proves J n ( f ) J n +1 ( f ). K n ( f ) K n +1 ( f ). Note that, inf I j,n ( f ) = min( inf I 2 j,n +1 ( f ) , inf I 2 j +1 ,n +1 ( f )) Thus, 1 2 n inf I j,n ( f ) = 1 2 n +1 inf I j,n ( f ) + 1 2 n +1 sup I j,n ( f ) (3) 1 2 n +1 inf I 2 j,n +1 ( f ) + 1 2 n +1 inf I 2 j +1 ,n +1 ( f ) (4) Summing the inequalities given by(4) over j = 0 , , 2 n 1 proves K n ( f ) K n +1 ( f ). K n ( f ) J n ( f ). Note that, 1 2 n inf I j,n ( f ) 1 2 n sup I j,n ( f ) (5) 1 Summing the inequalities given by(5) over j = 0 , , 2 n 1 proves K n ( f ) J n ( f ). Problem 1b Suffices to show that > N n N | J n ( f ) K n ( f ) | < . So, let > 0 be given. By uniform continuity of f , > 0 such that: < | x y | < | f ( x ) f ( y ) | < (6) Choose N big enough such that 1 2 N < . Then, clearly for n N , length( I j,n ) = 1 2 n 1 2 N < . Since f is continous, f assumes its maximum and minimum on I j,n . Thus, there is x j,n I j,n such that f ( x j,n ) = sup I j,n ( f ) and y j,n I j,n such that f ( y j,n ) = sup I j,n ( f ). Note that, | x j,n y j,n | length( I j,n ) < By the choice of in (6), we must have: sup I j,n ( f ) inf I j,n ( f ) = | f ( x j,n ) f ( y j,n ) | (7) < (8) In particular, 1 2 n (sup I j,n ( f ) inf I j,n ( f )) < 2 n (9) Summing equation(9) over j = { , , 2 n 1 } , gives J n ( f ) K n ( f ) < . Thus, weve shown that for n N , | J n ( f ) K n ( f ) | = J n ( f ) K n ( f ) < ....
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Homework_Set_6_Solution - Hw 6 solutions 10/31/2005 Problem...

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