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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?

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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 $I=(a,b)$.

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