# Homework Help: Another uniform convergence question

1. Feb 14, 2010

### Kate2010

1. The problem statement, all variables and given/known data

h:[0,1] -> R is continuous
Prove that t(x) = $$\sum$$$$^{infinity}_{n=0}$$ 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. Feb 14, 2010

### owlpride

Use the extreme value theorem to bound h(x^n), and then find small bounds for x^n h(x^n).

3. Feb 14, 2010

### rsa58

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.