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Prove the sequence converges uniformly

  1. Mar 28, 2009 #1
    1. The problem statement, all variables and given/known data
    Let f_n be a sequence of function whcih converges pointwise on [0,1] where each one is Lipschitz with the same constant C. Prove that the sequence converges uniformly.

    2. Relevant equations

    A function is called Lipschitz with Lipschitz constant C if |f(x)-f(y)| <= C|x-y| for all x,y in its domain.

    Let f_n be a sequence of functions defined on a set S. f is the pointwise limit of f_n if for all t in S lim n to infinity f_n(t) = f(t)


    3. The attempt at a solution
    I know that somehow I must show if for all epsilon > 0 there exists N in naturals such that sup|f_n(t) -f(t)| < epsilon if n>N for all t in [0,1]
    Or show the Uniform Cauchy Criterion holds : for all epsilon > 0 there exists N in naturals such that |f_n(t) -f_m(t)| < epsilon for all m,n > t for all t in [0,1].
     
  2. jcsd
  3. Mar 29, 2009 #2
    Go for the uniform Cauchy, use the triangle inequality and the fact that |x-y| could at most be 1.
     
  4. Mar 29, 2009 #3
    Hint: Pointwise convergence implies uniform convergence on any finite set of points. Since [0,1] is compact, you can choose points x1,...,xk such that the distances between consecutive points is arbitrarily small.
     
  5. Mar 29, 2009 #4
    I've tried using Cauchy, but I just seem to end back with a term I started with.
    Here's what I tried.
    |f_n(x) -f_m(x)| = |f_n(x) + f_n(y) + f_n(y) - f_m(x)| <= |f_n(x) + f_n(y)| + |f_n(y) - f_m(x)| <= C|x-y| + |f_n(y) - f_m(x)| = C|x-y| + |f_n(y) -f_n(x) + f_n(x) - f_m(x)| <=
    2C|x-y| + |f_n(x) - f_m(x)|

    As for yyat's hint, I don't believe that is true. The sequence of funtions could converge to a discontinuouse f(x) which would mean there could not be uniform convergence.
    If we knew Pointwise convergence implies uniform convergence on any finite set of points then we would not need the fact that the functions are Lipschitz.
     
  6. Mar 29, 2009 #5
    Any function defined on a finite set of points is continuous.

    Why? You want to prove uniform convergence on [0,1], which is not a finite set. The Lipschitz continuity is crucial here.
     
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