Proposition 1

A non-empty subset $A$ of $\mathbf{R}$ is an interval if and only if, whenever $a$ and $b$ are any two points of $A$ such that $a < b$, the closed interval $[a, b]$ is contained in $A$.

The condition is clearly necessary. Conversely, suppose that it is satisfied. If $A$ is neither bounded above nor below it must be the whole of $\mathbf{R}$, for if $x$ is any point of $\mathbf{R}$ there are then two points $a, b$ of $A$ such that $a < x < b$. If $A$ is bounded above but not below, let $k$ be its least upper bound; then for any $x < k$ there exist $a$ and $b$ in $A$ such that $a < x < b \leqslant k$, hence $x \in A$, and therefore $A$ can only be one of the two intervals $]\leftarrow, k], ]\leftarrow, k[$. The argument is similar in the other cases.