* Let $C([a, b])$ be the space of continuous functions $f : [a, b] \rightarrow \mathbb{R}$ where $[a, b] \subset \mathbb{R}$. Given any function $f \in C([a, b])$ define its moment sequence $f_n \in \mathbb{R}$ for $n = 0, 1, 2, \dots$ by
$f_n = \int_a^b f(x)x^n dx$.
Show that if $f, g \in C([0, 1])$ have the same moment sequence ($f_n = g_n$ for all $n = 0, 1, 2, \dots$) then $f = g$ on $[a, b]$.
[HINT: Let $h = f - g$ and $p$ be a polynomial. What is $\int_a^b h(x)p(x)dx$? ]
