Given and for all , show that

Singapore Mathematical Olympiad 2010

Inequalities

Easy

Inequalities by BJ Venkatachala

Do you really need a hint? Try it first!

Use induction.

Given the inequality for , the inequality for follows if .

Note that, as , the hypothesis implies that for all .

From hint 3, . Repeating this procedure, we eventually arrive at .
QED

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