If $I$ and $J$ are bounded intervals, then the intersection $I\cap J$ is also a bounded interval.
MY ATTEMPT
Since $I$ and $J$ are bounded, we conclude that $I\subseteq[-M,M]$ and $J\subseteq[-N,N]$.
Without loss of generality, we may assume that $M > N$. Therefore $I\cap J\subseteq[-M,M]$.
Now it remains to prove that $I\cap J$ is an interval. There are three possibilities:
(a) $|I\cap J| = 0$
(b) $|I\cap J| = 1$
(c) $|I\cap J| > 1$.
In the first case, $I\cap J$ is the empty set, which is an interval.
In the second case, $I\cap J = \{a\}$, which is an interval as well.
In the third case, there are at least two points in $I\cap J$ (in fact, there are infinitely many).
If $x\in I\cap J$ and $y\in I\cap J$ such that $x < y$, we conclude that $x\in I$ and $y\in I$. Since $I$ is an interval, it results that $[x,y]\subseteq I$. Similar reasoning leads to the conclusion $[x,y]\subseteq J$. Hence $[x,y]\subseteq I\cap J$, and the result follows.
Can someone please double-check my reasoning?