# Sequence Cauchy in L2?

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