Any help understanding this Lemma is Appreciated

[Mining_Engr]

The lemma and proof are attached, how does the hypothesis imply that ∫h(u)*p(u)du = 0. This is where I am hung up. I don't see how the hypothesis implies this, please help!

Thanks in advance,

Mining_Engr

 

[Office_Shredder]

p(u) is a polynomial, say
$$p(u) = \sum_{i=0}^n \alpha_i u^i$$

then
$$\int h(u) p(u) du = \int h(u) \sum_{i=0}^n \alpha_i u^i du = \sum_{i=0}^n \alpha_i \int h(u) u^i du$$
and all those integrals at the end are zero so the whole sum is zero