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

Question

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

How do I show this?

I made a plot, and it looks true:

t.b. · Accepted Answer · 2011-04-16 22:20:11Z

up vote 8 down vote accepted

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

answered Apr 16 '11 at 22:20

t.b.
43.5k3112196

I of course assume that you're only considering $x \gt 0$. – t.b. Apr 16 '11 at 22:32

mac · Answer 2 · 2011-04-16 22:21:15Z

up vote 7 down vote

Divide both sides by $\lceil \sqrt x\rceil$.

answered Apr 16 '11 at 22:21

mac
1,528513

2 Answers