Triangle Inequality Issue Suppose that $\left| a \right| \leq 1$ and $\left| b \right| \leq 1$, is there a nice way, other than a proof by cases, to show that $$\left| \left| a \right|^n - \left| b \right|^n \right| \leq 1?$$ I'm obviously aware of the triangle inequality $\left| \left| a \right| - \left| b \right| \right| \leq \left| a - b \right|$, but this doesn't help me out. 

From $|a|^n \le 1$ we get $|a|^n \le 1+|b|^n$, hence $|a|^n-|b|^n \le 1$
In the same way we show: $|b|^n-|a|^n \le 1$
