Proof of uniqueness of Laplace Transform

Question

I'm studying proof of the uniqueness of the Laplace Transform and I have some problems with understanding it.

Here is the proof.

Why can we change $u^{n}$ for polynomial $p(u)$? What does it mean?

Answer 1

Hint:

Let $p(u)=\sum_{n=0}^k a_n u^n.$ Now use $(1)$ of the proof and linearity of integral.