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$x\le\lceil\sqrt{x}\rceil\left\lceil\frac{x}{\lceil\sqrt{x}\rceil}\right\rceil$

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I made a plot, and it looks true:

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2 Answers

up vote 8 down vote accepted

Hint: $\frac{x}{\lceil\sqrt{x}\rceil} \leq \lceil\frac{x}{\lceil\sqrt{x}\rceil}\rceil$.

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I of course assume that you're only considering $x \gt 0$. –  t.b. Apr 16 '11 at 22:32
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Divide both sides by $\lceil \sqrt x\rceil$.

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