Unformatted text preview: ( f n ) , a real valued sequence of functions, converges uniformly to f . (a) (8 pts) Prove that if g : R → R is a Lipshitz function, then the sequence g ( f n )) converges uniformly to g ( f ) . (b) (7 pts) Prove that if ( f n ) above are bounded functions, then f is bounded. 5. Let f n ( x ) = n c x (1-x 2 ) n , for all x ∈ R and n ∈ N . (a) (5 pts) Prove that ( f n ) converges pointwise in [0 , 1] for all c ∈ R . (b) (10 pts) Determine those c for which the convergence is also uniform in [0 , 1] . 6. (20 pts) Problem 4, page 165 in Rudin ....
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- Fall '10
- Calculus, 5 pts, 8 pts, 10 pts, 7 pts