TITLE: $x\le\lceil\sqrt{x}\rceil\left\lceil\frac{x}{\lceil\sqrt{x}\rceil}\right\rceil$?
QUESTION [6 upvotes]: $x\le\lceil\sqrt{x}\rceil\left\lceil\frac{x}{\lceil\sqrt{x}\rceil}\right\rceil$
How do I show this?
I made a plot, and it looks true:

REPLY [8 votes]: Hint: $\frac{x}{\lceil\sqrt{x}\rceil} \leq \lceil\frac{x}{\lceil\sqrt{x}\rceil}\rceil$.

REPLY [7 votes]: Divide both sides by $\lceil \sqrt x\rceil$.