Numerical-range endpoint approximate-eigenvalue theorem
= Numerical-range endpoint approximate-eigenvalue theorem
{title2=$\inf W(T),\sup W(T)\in\sigma_{\mathrm{ap}}(T)$}
For bounded <self-adjoint> $T$, put $B=\beta I-T\geq0$ with $\beta=\sup W(T)$. <Unit vectors> with $\langle Bu_n,u_n\rangle\to0$ satisfy $\|Bu_n\|^2\leq\|B\|\langle Bu_n,u_n\rangle$ by positive-form <Cauchy-Schwarz inequality>. Thus $\beta$ is an <approximate eigenvalue>. Applying the same argument to $T-\alpha I$ proves the result at $\alpha=\inf W(T)$.