Residual extrema when response extrema are attained ($L-M\le e_{\min }\le L-m,U-M\le e_{\max }\le U-m$)

For a finite sample, let $L={\min }_i{y}_i$, $U={\max }_i{y}_i$, $m={\min }_i{\hat{y}}_i$ and $M={\max }_i{\hat{y}}_i$. The regression residuals are $e_i=y_i-{\hat{y}}_i$. Every residual is at least $L-M$ and at most $U-m$. At an observation attaining $y_i=L$, its residual is at most $L-m$, so $e_{\min }\le L-m$. At an observation attaining $y_i=U$, its residual is at least $U-M$, so $e_{\max }\ge U-M$. These observations prove all four bounds. Merely knowing theoretical response limits, without attainment, would not justify the two inner bounds.