
Sequence Cauchy in L2?
Is the sequence {sin(nx)} Cauchy in L2 over [0,pi]? Why or why not?
Under the L2 norm, this would amount to the integral of sin(nx)sin(mx)^2 from 0 to pi being bounded by small epsilon for all n,m > N, right?
I can't figure this guy out. Thanks for any help in advance.

$\displaystyle \{ \sin (nx) \}$ is an orthogonal set in $\displaystyle L^2([0, \pi ])$ (prove it) and so remember that if $\displaystyle u$ and $\displaystyle v$ are orthogonal vectors $\displaystyle \Vert u + v \Vert ^2 = \Vert uv \Vert ^2 = \Vert u \Vert ^2 + \Vert v \Vert ^2$ (Because $\displaystyle L^2([0, \pi ])$ is a Hilbert sapce).