I have been given the following statement by a professor:
Let $m\leq x\leq M$ and define $K=\max(|m|,|M|)$. Then $|x|\leq K$.
Now I can clearly see that this is true, and working through case-by-case (i.e. $m>0$ and $M>0$, $m<0$ and $M>0$ with $|m|<|M|$ etc) I can also prove it is true for each case. But I feel that there must be a much more elegant proof than the one I have. Can anyone think of one or know one?