1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted