# Equivalent definitions of interval

I see two definitions of interval in different sources.

First one is:

A (real) interval is a subset I of the real numbers such that:

$$\forall x, y \in I: \forall z \in \mathbb{R} : ({x < z < y \implies z \in I})$$

Second one is:

Let $$a, b \in \mathbb{R}^*$$ be extended real numbers. We define the closed interval $$[a, b]$$ by $$[a, b] : = \{x \in \mathbb{R}^* : a \le x \le b\}$$. The half-open intervals and the open intervals are defined in a similar fashion.

I can see that Second definition implies first definition. But i can not see the other direction. How can we prove that?

Just consider the infimum and supremum of the given interval $$I$$.
Specifically, let $$a=\inf I\in\Bbb R^*$$ and $$b=\sup I\in\Bbb R^*$$.
Now if $$a,b\in I$$ we have $$I=[a,b]$$,
If $$a\in I$$ but $$b\notin I$$, then $$I=[a,b)$$,
If $$a\notin I,\ b\in I$$, then $$I=(a,b]$$,
And if $$a,b\notin I$$, then $$I=(a,b)$$.