Interval in the Real Line

A subset $I$ of $\mathbb{R}$ is called an interval if for all $x,y,z\in\mathbb{R}$, whenever $x,z\in I$ and $x\le y\le z$, one has $y\in I$. If $a,b\in\mathbb{R}$ satisfy $a\le b$, the closed interval $[a,b]$ is the set $\{x\in\mathbb{R}: a\le x\le b\}$.