The Sugar-Water Inequality at Work

Problem

Dissolve $a$ grams of sugar in $b$ grams of sugar water ( $b > a > 0$ ). Adding $m > 0$ grams of sugar makes the water sweeter — which is the everyday content of the inequality

\frac{a+m}{b+m} > \frac{a}{b}.

Use this sugar-water inequality to prove that for every positive integer $n$ ,

\left(1 - \frac12\right)\left(1 - \frac14\right)\left(1 - \frac16\right)\cdots\left(1 - \frac{1}{2n}\right) < \sqrt{\frac{1}{2n+1}}.

Answer

Solution

Difficulty	5/10
Topics	inequality

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