Another uniform convergence question

[Kate2010]

Homework Statement

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

Homework Equations

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.

 

[owlpride]

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

 

[rsa58]

h is continuous (it is a continuous 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.