I've seen the full proof of the Triangle Inequality
\begin{equation*}
|x+y|\le|x|+|y|.
\end{equation*}
However, I haven't seen the proof of the reverse triangle inequality:
\begin{equation*}
||x|-|y||\le|x-y|.
\end{equation*}
Would you please prove this using only the Triangle Inequality above?
Thank you very much.
Best Answer
$$|x| + |y -x| \ge |x + y -x| = |y|$$
$$|y| + |x -y| \ge |y + x -y| = |x|$$
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:
$$|x-y| \ge \bigl||x|-|y|\bigr|.$$