How would you prove that $mLet $M,m,a,b\in\Bbb R$. How would you prove that $m<a,b<M$ then $|a-b|<M-m$? This little question came from a part of an exercise in analysis. Of course, this is quite simple to see if we draw these points on a number line. But how can we show this rigorously and simply?
PS: By triangle inequality, we can show that $|a-b|\leq|a-m|+|m-M|+|M-b|$, but this does not show that $|a-b|<M-m$.
 A: We have $$a - b < M - m$$ and $$-(M - m) = m - M < a - b$$ Hence together $$|a - b| < M-m$$ By using that $$|x| < c \quad \Leftrightarrow \quad -c < x < c$$ for $x \in \mathbb{R}$ and $c > 0$.
A: Pf I: (A super easy proof)
Note that $\begin{cases}m<a<M\\m<b<M\end{cases}$, then $a-b<M-m$, since $M-m$ is the greatest possible value minus the lowest possible value. By the same reason we also  have $b-a<M-m$.
Pf II:
No matter $a$ or $b$ is the bigger, since $a<M$ and $m<b$, then $-b<-m$,
Add these two expression, we have $a+(-b)<M+(-m)$, namely $a-b<M-m$.
Since the choice of $a,b$ is arbitrary, then $b-a<M-m$ also holds.
Hence no matter $|a-b|=a-b$ or $b-a$, we have $|a-b|<M-m$.
A: Inequalities of real numbers like the one in your question follow easily from the properties of order relations of real numbers, but they are not obvious / self evident unless visualized on the number line. Also note that using number line is rigorous enough because of the assumption that to each point on the number line corresponds a unique real number and vice versa and the linear order from left to right on the number line is equivalent to the ordering of real numbers.
But in case you wish to avoid the number line here is a simple argument. Without any loss of generality we can assume $a<b$ because the case $a=b$ is trivial and the case $a>b$ can be obtained by reversing the roles of $a, b$. We then have $$m<a<b<M$$ and clearly $$M-m=(M-b) + b -  a + (a-m) $$ The terms in parentheses on the right side of the above equation are positive and hence $M-m>b-a=|a-b|$. The technique of splitting an expression into multiple terms by adding and subtracting the same quantity is heavily used in analysis mainly to establish inequalities like these.
