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Let $A$, $\overline{A}$, $\underline{A}$ and $B$ be real numbers, and let $\underline{A}\le A\le \overline{A}$. Is the following correct? $$ |A-B| = \max\{A-B, B-A\} \le \max\{\overline{A}-B, B-\underline{A}\} $$

Apologies for the elementary question.

user52227
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    Well, $$\underline A\leqslant A\implies -\underline A\geqslant-A\implies B-\underline A\geqslant B-A\\overline A\geqslant A\implies \overline A-B\geqslant A-B$$ –  Mar 02 '16 at 14:35
  • Thanks @G.Sassatelli So you're saying that I am correct, right? – user52227 Mar 03 '16 at 09:15
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    Yes: the last inequality comes from the fact that each number in the first $\max{\ }$ is smaller that some number in the second $\max{\ }$ –  Mar 03 '16 at 14:21

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