Memorize inequalities about floor function

Question

Let $n \in \mathbb{Z}$. Prove that: 1.$\lfloor x \rfloor \le n \to x \lt n+1$

2.$ \lfloor x \rfloor \lt n \to x \lt n$

3.$ \lfloor x \rfloor \ge n \to x \ge n$

4.$\lfloor x \rfloor \gt n \to x \ge n+1$

Since I'm not able to memorizing these inequalities , looking for a method based on reasoning in order to stick in mind .

Answer 1

$[x]$ it's an integer number for which $$[x]\leq x<[x]+1.$$

It's obvious if $f(x)=[x]$ then $f$ is a function and from here we can get all properties.

For example, if $[x]\leq n$ then since $[x]>x-1$, we obtain $x-1<n$ or $x<n+1$.