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

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:

score 8 · Accepted Answer · answered Apr 16 '11 at 22:20

8

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.

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

score 7 · Answer 2 · answered Apr 16 '11 at 22:21

7

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

answered Apr 16 '11 at 22:21

mac

2 Answers2