proof of Bessel inequality

Let

$$r_n=x-\sum _{k=1}^n⟨x,e_k⟩\cdot e_k.$$

Then for $j=1,\mathrm{\dots },n$,

$⟨r_n,e_j⟩$$=⟨x,e_j⟩-{\displaystyle \sum _{k=1}^n}⟨⟨x,e_k⟩\cdot e_k,e_j⟩$(1)

$=⟨x,e_j⟩-⟨x,e_j⟩⟨e_j,e_j⟩=0$(2)

so $e_1,\mathrm{\dots },e_n,r_n$ is an orthogonal series.

Computing norms, we see that

$${\parallel x\parallel }^2={\parallel r_n+\sum _{k=1}^n⟨x,e_k⟩\cdot e_k\parallel }^2={\parallel r_n\parallel }^2+\sum _{k=1}^n{\left|⟨x,e_k⟩\right|}^2\ge \sum _{k=1}^n{\left|⟨x,e_k⟩\right|}^2.$$

So the series

$$\sum _{k=1}^{\mathrm{\infty }}{\left|⟨x,e_k⟩\right|}^2$$

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