Cover of Interval By Closed Intervals is not Pairwise Disjoint

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Theorem

Let $\left[{a \,.\,.\, b}\right]$ be a closed interval in $\R$.



Let $\mathcal J$ be a set of two or more closed intervals contained in $\left[{a \,.\,.\, b}\right]$ such that $\displaystyle \bigcup \mathcal J = \left[{a \,.\,.\, b}\right]$.

Then the intervals in $\mathcal J$ are not pairwise disjoint.


Proof

Suppose that the intervals of $\mathcal J$ are pairwise disjoint.

Let $I = \left[{p \,.\,.\, q}\right]$ be the unique interval of $\mathcal J$ such that $a \in I$.

Let $J = \left[{r \,.\,.\, s}\right]$ be the unique interval of $\mathcal J$ containing the least real number not in $I$.

These choices are possible since $\mathcal J$ has at least two elements, and they are supposed disjoint.

If $q = r$ then $I \cap J \ne \varnothing$, a contradiction.

If $r < q$ then $I \cap J \ne \varnothing$, a contradiction.

If $r > q$ then there is some real number with $q < \alpha < r$.

Therefore $\alpha \notin I$.

Since $J$ contains the least real number not in $I$ it follows that there is $\beta \in J$ with $\beta < \alpha$.

But we also have that $\alpha < r \le \beta$ for all $\beta \in J$, a contradiction.

This exhausts all the possibilities, and we conclude that the intervals of $\mathcal J$ are not[[Definition:Pairwise Disjoint|pairwise disjoint].

$\blacksquare$