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

How do I show this?

I made a plot, and it looks true:

2 Answers 2

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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 $x \gt 0$.2011-04-16
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Divide both sides by $\lceil \sqrt x\rceil$.