- #1

Hitman2-2

## Homework Statement

Let

*f*be a continuous function on [0,1]. Prove that if

[tex]

\int_{0}^{1} x^n f(x) dx = 0

[/tex]

for all even natural numbers

*n*, then

*f(x)*= 0 for all [tex]x \in [0,1] [/tex].

## Homework Equations

## The Attempt at a Solution

I'm pretty much stuck on this problem. All I know is that by the Weierstrass Approximation Theorem, there exists a sequence of polynomials

*P*

_{n}that converges uniformly to

*f*, so

[tex]

\lim_{n\rightarrow 0} \int_{0}^{1} P_n f(x) dx = \int_{0}^{1} f(x) \lim_{n\rightarrow 0} P_n dx

= \int_{0}^{1} f^2(x) dx

[/tex]

but that's about as far as I can get.