Inequality involving the min function I'm trying to prove the following inequality: 
$$
\left|y_{1}\land x_{1}-y_{2}\land x_{2}\right|\leq\left|y_{1}-y_{2}\right|+\left|x_{1}-x_{2}\right|,
$$
where $x\land y=\mbox{min}(x,y)$. By considering different cases
it seems to hold but I haven't been able to find a simple way of proving
it. I'd appreciate any suggestions. 

Hint: Consider the four cases:


*

*$y_1<x_1$ and $y_2<x_2$

*$y_1<x_1$ and $y_2\geq x_2$

*$y_1\geq x_1$ and $y_2<x_2$

*$y_1\geq x_1$ and $y_2\geq x_2$
By symmetry, you only need to consider the first two cases.


*

*The first case is simple because the LHS is a term on the RHS.

*For the second case, either $y_1<x_2<y_2$ so that $|y_1-y_2|=|y_1-x_2|+|x_2-y_2|$ or $x_2<y_1<x_1$ so $|x_2-x_1|=|x_2-y_1|+|y_1-x_1|$.  In either case, we get that $|y_1-x_2|$ is less than or equal to a term on the RHS.
