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Sequence of Functions Converges to Continuous Function Implies Convergence Is Uniform

  1. Jan 13, 2009 #1
    The problem statement, all variables and given/known data
    Say we have a sequence of functions {fn: [a,b] -> R} such that each fn is nondecreasing. Suppose that {fn} converges point-wise to f. Prove that if f is continuous, then {fn} converges uniformly to f.

    The attempt at a solution
    Fix an x in [a,b] and let e > 0. Then we can find a d such that if |y - x| < d, then |f(y) - f(x)| < e. Now fix a y such that |y - x| < d. Then there is an N such that |f(y) - fn(y)| < e for all n > N. And so we have

    |f(x) - fn(x)| <= |f(x) - f(y)| + |f(y) - fn(y)| + |fn(y) - fn(x)|.

    The first two terms on the RHS can be made arbitrarily small, but not the last one. I haven't used the fact that fn or that f is nondecreasing, but I don't understand how this would come into play. Any tips?
     
  2. jcsd
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