Floor function inequality

[sweatingbear]

Let [x] be the floor function i.e. it produces the integral part of x. So for example if x = 1.5 then [x] = 1. I recently saw the claim

\(\displaystyle [x] \geq x - 1\)

The strict part of the inequality makes perfect sense, but when does equality occur? Does it even occur at all? I have not been able to find an example. Maybe the claim is false?

 

[Klaas van Aarsen]

Let [x] be the floor function i.e. it produces the integral part of x. So for example if x = 1.5 then [x] = 1. I recently saw the claim

\(\displaystyle [x] \geq x - 1\)

The strict part of the inequality makes perfect sense, but when does equality occur? Does it even occur at all? I have not been able to find an example. Maybe the claim is false?

Hi sweatingbear!

You are right that equality cannot occur.
However, the claim is still true.
Consider that:
$$[x] > x - 1 \quad\Rightarrow\quad [x] \geq x - 1$$
It's just a weaker statement. Still true though.