I am stuck trying to mathematically prove the first statement that was made in the top answer to this question involving minimizing the sum of absolute differences between pairings of boys' and girls' heights.
"If two boys have heights $b_1\le b_2$ and two girls have heights $g_1\le g_2$ and if you have a matching where $b_1$ is paired with $g_2$ and $b_2$ is paired with $g_1$, then you can find at least as good of a matching by instead matching $b_1$ with $g_1$ and $b_2$ with $g_2$."
Basically, some way to prove that given $b_1\le b_2$ and $g_1\le g_2$, the following holds: $$|g_1-b_1| + |g_2 - b_2| \le |g_1-b_2| + |g_2 - b_1|$$
I have tried looking at things like the triangle inequality and using other properties of absolute value but don't seem to be getting anywhere. Would anybody be able to explain this?