Unformatted text preview: M,M ], show that the sequence { g ◦ f n } converges uniformly to g ◦ f on A . Fix ε > 0. If g is continuous on the compact interval [M,M ], it is actually uniformly continuous. Thus, there is an η > 0 so that if x,y ∈ [M,M ] with  xy  < η , then  g ( x )g ( y )  < ε . Moreover, since f n → f uniformly, there is an N ∈ N so that  f n ( x )f ( x )  < η for all x ∈ A provided n ≥ N . Combining these, we have that for x ∈ A and n ≥ N , then  g ( f n ( x ))g ( f ( x ))  < ε which shows the desired uniform convergence. 1...
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This note was uploaded on 07/15/2010 for the course MATH 521 taught by Professor Metcalfe during the Spring '10 term at University of North Carolina Wilmington.
 Spring '10
 MetCalfe
 Calculus

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