22 lines
458 B
TeX
22 lines
458 B
TeX
\beginsection{3.5}
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(a) Show that $|b|\le a$ if and only if $-a\le b\le a$.
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\medskip
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First prove the implication.
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Let $|b|\le a$.
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Then by Theorem 3.2, $-a\le-|b|$.
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By $-|b|\le b\le|b|$ we have
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$$-a\le-|b|\le b\le|b|\le a$$
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Hence $|b|\le a$ implies $-a\le b\le a$.
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\medskip
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Now prove the converse.
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By hypothesis we have
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$$b\le a\eqno(1)$$
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Also by hypothesis we have $-a\le b$ which implies
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$$-b\le a\eqno(2)$$
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Therefore by (1) and (2) we have
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$$|b|\le a$$
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