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**Homework Statement**

Suppose f: [0,1] --> R is continuous and

[tex]\int_0^1 f(x) x^n \, dx = 0[/tex]

for all n = 0, 1, ... Prove that f(x) = 0.

**The attempt at a solution**

There's a hint that says: Prove that

[tex]\int_0^1 (f(x))^2 \, dx = 0.[/tex]

I don't know how to prove this hint and I don't know how that would help in determining what f is. Any tips?