$|a| - |b - a| \leq |b|$ How would you prove that $|a| - |b - a| \leq |b|$ for all $a, b \in \mathbb R$? I've tried several methods so far like the triangle inequality, but nothing seems to work. Can someone please help?
Thanks in advance.

First, write: 
\begin{align*}
|a| = |a + 0| = |(a - b) + b|.
\end{align*}
Using the triangle inequality:
\begin{align*}
|a| \leq |a-b| + |b|
\end{align*}
Noting that $|a - b| = |b-a|$, write: 
\begin{align*}
|a| \leq |b- a| + |b|.
\end{align*}
Subtracting $|b - a|$ gives
\begin{align*}
|a| - |b-a| \leq |b|.
\end{align*}
