proof of the Bessel inequality (idea)

See the Bessel inequality for what we'll prove.

Since we're approximating x, let's write down the exact approximation and error term:

y_n = ∑_i=1ⁿ⟨x,e_i⟩e_i
r_n = x - y_n

Now a bit of simple magic: r_n is x without its projections on e₁,...,e_n, so it should be orthogonal to all of them

Indeed, for j=1,...,n, we have that

⟨r_n,e_j⟩ = ⟨x,e_j⟩ - ∑_i=1ⁿ⟨⟨x,e_i⟩e_i,e_j⟩ =

(All elements of the sum disappear except for i=j, since ⟨e_i,e_j⟩=0...)

⟨x,e_j⟩ - ⟨x,e_j = 0.

Thus we have a shorter orthogonal sequence of elements e₁,...,e_n,r_n; in particular, r_n⊥y_n

But (applying what is essentially the Pythagorean theorem...) we know that

||x||² = ||r_n+y_n||² = ⟨r_n+y_n,r_n+y_n⟩ = ||r_n||²+||y_n||²

The first term ||r_n||² is nonnegative, so

||y_n||² ≤ ||x||².

Computing ||y_n||²=∑_i=1ⁿ|⟨x,e_i⟩|² yields the inequality.