Reverse Triangle Inequality Proof

$$|x| + |y -x| \ge |x + y -x| = |y|$$

$$|y| + |x -y| \ge |y + x -y| = |x|$$

Move $|x|$ to the right hand side in the first inequality and $|y|$ to the right hand side in the second inequality. We get

$$|y -x| \ge |y| - |x|$$

$$|x -y| \ge |x| -|y|.$$

From absolute value properties, we know that $|y-x| = |x-y|,$ and if $t \ge a$ and $t \ge −a$ then $t \ge |a|$.

Combining these two facts together, we get the reverse triangle inequality:

$$|x-y| \ge \bigl||x|-|y|\bigr|.$$

WLOG, consider $|x|\ge |y|$. Hence: $$||x|-|y||=||x-y+y|-|y||\le ||x-y|+|y|-|y||=||x-y||=|x-y|.$$

Explicitly, we have \begin{align} \bigl||x|-|y|\bigr| =& \left\{ \begin{array}{ll} |x-y|=x-y,&x\geq{}y\geq0\\ |x-y|=-x+y=-(x-y),&y\geq{}x\geq0\\ |-x-y|=x+y\leq-x+y=-(x-y),&y\geq-x\geq0\\ |-x-y|=-x-y\leq-x+y=-(x-y),&-x\geq{}y\geq0\\ |-x+y|=-x+y=-(x-y),&-x\geq-y\geq0\\ |-x+y|=x-y,&-y\geq-x\geq0\\ |-x+y|=-x-y\leq{}x-y,&-y\geq{}x\geq0\\ |-x+y|=x+y\leq{}x-y,&x\geq-y\geq0 \end{array} \right\}\nonumber\\ =&|x-y|.\nonumber \end{align}