= Residual extrema when response extrema are attained
{title2=$L-M\le e_{\min}\le L-m,\quad U-M\le e_{\max}\le U-m$}
For a finite sample, let $L=\min_i y_i$, $U=\max_i y_i$, $m=\min_i\widehat y_i$ and $M=\max_i\widehat y_i$. The <regression residuals> are $e_i=y_i-\widehat y_i$. 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.
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