Property involving Maximum and Absolute Value function

Question

I am trying to show $$ |max\{a,b\} - max\{c,d\}| \leq max\{|a-c|,|b-d|\}. $$ There may be a slicker way to prove this but I considered the following cases. The trivial cases are when $a \geq b$ and $c \geq d$ or $b \geq a$ and $d \geq c$. Because in the first case $$|max\{a,b\} - max\{c,d\}| = |a - c| \leq max\{|a-c|,|b-d|\}$$ and similarly for the second. The case where $a \geq b$ and $d \geq c$ is where I am stumped. We arrive at $|max\{a,b\} - max\{c,d\}| = |a - d|$, but I'm struggling to show $|a - d| \leq |a - c|$ or $|a - d| \leq |b - d|$. The other non-trivial case is similar.

Answer 1

Notice for this case we have $$b - d \leq a - d \leq a - c. $$ So, if $a - d \leq 0$ then $|a - d| \leq |b - d|$ and if $a - d \geq 0$ then $|a - d| \leq |a - c|$. So, in either case $|a - d| \leq max\{|a - c|,|b - d|\}.$