proof of Bessel inequality

Let

r_{n}=x-\sum_{k=1}^{n}\left\langle x,e_{k}\right\rangle\cdot e_{k}.

Then for $j=1,\ldots,n$ ,

	$\displaystyle\left\langle r_{n},e_{j}\right\rangle$	$\displaystyle=\left\langle x,e_{j}\right\rangle-\sum_{k=1}^{n}\left\langle% \left\langle x,e_{k}\right\rangle\cdot e_{k},e_{j}\right\rangle$			(1)
		$\displaystyle=\left\langle x,e_{j}\right\rangle-\left\langle x,e_{j}\right% \rangle\left\langle e_{j},e_{j}\right\rangle=0$			(2)

so $e_{1},\ldots,e_{n},r_{n}$ is an orthogonal series.

Computing norms, we see that

\left\|x\right\|^{2}=\left\|r_{n}+\sum_{k=1}^{n}\left\langle x,e_{k}\right% \rangle\cdot e_{k}\right\|^{2}=\left\|r_{n}\right\|^{2}+\sum_{k=1}^{n}\left|% \left\langle x,e_{k}\right\rangle\right|^{2}\geq\sum_{k=1}^{n}\left|\left% \langle x,e_{k}\right\rangle\right|^{2}.

So the series

\sum_{k=1}^{\infty}\left|\left\langle x,e_{k}\right\rangle\right|^{2}

converges and is bounded by $\left\|x\right\|^{2}$ , as required.