# Greatest Integer Function

• Mar 25th 2006, 06:21 PM
ThePerfectHacker
Greatest Integer Function
Can you prove that for any real number $x$ there exists as unique integer $n$ such as,
$x-1\leq n\leq x$
thus, the function $f(x)=[\x x\x ]$ is well-defined.
• Mar 28th 2006, 06:12 AM
Rebesques
Well... there certainly exists one such number, n(x); Or else, the naturals would be bounded in the reals, something the Archimedean Property denies.

So there is one, at least. The set {n(x): n(x)-1< x <n(x)} must have a least element, by the Well-Ordering Property of the naturals. This least element, is exactly [x].

Personally, I would not bother myself so much :p just take x, and kill its integer part!