$f: [ 4, + \infty) \to \mathbb R$
$f(x)= 1/Ax$
Prove: $| f(x) - f(y) | \le 1/16 $
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2026-04-13 02:52:52.1776048772
prove inequality $| f(x) - f(y) | < 1/16$
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Suppose $A \ge 8$, then $|f(x)-f(y)| = \left|\dfrac{1}{Ax}-\dfrac{1}{Ay}\right| = \dfrac{|x-y|}{Axy} \le \dfrac{|x|+|y|}{xyA} = \dfrac{1}{A}\left(\dfrac{1}{x}+\dfrac{1}{y}\right) \le \dfrac{1}{A}\left(\dfrac{1}{4}+\dfrac{1}{4}\right) =\dfrac{1}{2A} \le \dfrac{1}{16}$, as claimed.