 Solution

The residual $x-P_{n}x$ is orthogonal to $P_{n}x\in U_n$. Thus the Pythagorean theorem gives $\Vert x{\Vert }^2=\Vert x-P_{n}x{\Vert }^2+\Vert P_{n}x{\Vert }^2$. Substitute part (ii) and rearrange:Nonnegativity of the left-hand side also gives the finite form of Bessel's inequality.