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Homework Help: Another uniform convergence question

  1. Feb 14, 2010 #1
    1. The problem statement, all variables and given/known data

    h:[0,1] -> R is continuous
    Prove that t(x) = [tex]\sum[/tex][tex]^{infinity}_{n=0}[/tex] xnh(xn) is uniformly convergent on [0,s] where 0<s<1

    2. Relevant equations



    3. The attempt at a solution

    I have the definition of h being continuous but after this I am pretty clueless about how to tackle this problem. I could use the Weierstrass M-test. I know the series xn converges uniformly on this interval as xn < sn but I don't know how to use the fact that h is continuous to find a sequence of real numbers that xnh(xn) is always less than.
     
  2. jcsd
  3. Feb 14, 2010 #2
    Use the extreme value theorem to bound h(x^n), and then find small bounds for x^n h(x^n).
     
  4. Feb 14, 2010 #3
    h is continous (it is a continous function of a polynomial which is continuous) on a compact set therefore h is bounded for x in the interval [0,s]. the sequence of h's for all n is uniformly bounded by a single M. as you said the s^n is a geometric series so it converges. hence uniform convergence.
     
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