Numerical-range endpoint approximate-eigenvalue theorem ($\inf W(T),\sup W(T)\in {\sigma }_{\mathrm{ap}}(T)$)

For bounded self-adjoint $T$, put $B=\beta I-T\ge 0$ with $\beta =\sup W(T)$. Unit vectors with $⟨B{u}_n,u_n⟩\to 0$ satisfy $\Vert B{u}_n{\Vert }^2\le \Vert B\Vert ⟨B{u}_n,u_n⟩$ by positive-form Cauchy-Schwarz inequality. Thus $\beta$ is an approximate eigenvalue. Applying the same argument to $T-\alpha I$ proves the result at $\alpha =\inf W(T)$.