Unformatted text preview: f is continuous, then at least one of the functions f n must be continuous. Problem 4: Let ( f n ) be a sequence of monotonically increasing functions which converges pointwise to f . Prove that f must also be monotonically increasing. Problem 5: Let f n : [ a;b ] ! R be a sequence of continuous functions, and ( x n ) be a sequence in [ a;b ] converging to x . If f n ! f uniformly, then prove that f n ( x n ) ! f ( x ). Problem 6: De±ne a sequence of functions f n : R ! R by: f n ( x ) = n X k =1 1 x 2 + k 2 Prove that f n converges uniformly on R ....
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- Fall '07