Prove $||a| - |b|| \leq |a - b|$ I'm trying to prove that $||a| - |b|| \leq |a - b|$. So far, by using the triangle inequality, I've got:
$$|a| = |\left(a - b\right) + b| \leq |a - b| + |b|$$
Subtracting $|b|$ from both sides yields,
$$|a| - |b| \leq |a - b|$$
The book I'm working from claims you can achieve this proof by considering just two cases: $|a| - |b| \geq 0$ and $|a| - |b| < 0$. The first case is pretty straightforward:
$$|a| - |b| \geq 0 \implies ||a| - |b|| = |a| - |b| \leq |a - b|$$
But I'm stuck on the case where $|a| - |b| < 0$
Cool, I think I got it (thanks for the hints!). So,
$$|b| - |a| \leq |b - a| = |a - b|$$
And when $|a| - |b| < 0$,
$$||a| - |b|| = -\left(|a| - |b|\right) = |b| - |a| \leq |a - b|$$

Hint: If $|a|-|b|<0$, rename $a$ to $b'$ and $b$ to $a'$.
