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CaptainBlack · #10 November 11th, 2007, 07:23 AM

Quote:

Originally Posted by Jhevon

i always have trouble proving things that seem to me to be obvious.

would it be wrong to do this?

$\int_0^1 x^n f(x)~dx = \int_0^1 x^n g(x)~dx$

$\Rightarrow \int_0^1 x^n f(x)~dx - \int_0^1 x^n g(x)~dx = 0$

$\Rightarrow \int_0^1 x^n [f(x) - g(x)]~dx = 0$

Obviously, the only way this integral can be zero for all x and all n is if $f(x) - g(x) = 0$ . the result follows immediately.

Now i can see where the problem here would be, the "obviously" part.

These proofs are thinly disguised use of the Stone-Weierstrass theorem.

RonL