Property of absolute value in the real numbers

Question

To prove that $ \lvert a-b \rvert \le c-d $ for $ a,b,c,d $ in the real numbers, what needs to be shown?

Is the fact that $a-b\le c-d$ enough? Or is there something more that needs to be shown?

Answer 1

$a - b \le c - d$ is not enough.

For example, $a = b = -999, c = d = 1$.

What you need to show is that $a - b \le c - d$ and $b - a \le c - d$. You need both.