Proving the reverse triangle inequality of the complex numbers
Solution 1:
You know that $|x| \le |x-y|+|y|$ and so $|x|-|y| \le |x-y|$. The same argument with $x,y$ switched gives $|y|-|x| \le |y-x| = |x-y|$. Hence $||x|-|y|| \le |x-y|$.
This is true for any norm, not just the modulus. The essential element here is the triangle inequality.