How combine two inequalities (complex numbers)? This is from a book. I don't understand how the inequalities are combined to one inequality. Are they added/subtracted?

$z_1$ and $z_2$ are complex numbers. We have the inequalities
  $$\lvert z_2\rvert -\lvert z_1\rvert \leq \lvert z_2-z_1\rvert $$ 
  $$\lvert z_1\rvert -\lvert z_2\rvert \leq \lvert z_1-z_2\rvert $$
  Combining these inequalities gives
  $$\lvert \lvert z_1\rvert -\lvert z_2\rvert \rvert \leq \lvert z_1-z_2\rvert$$

Attempt
If I add them:
\begin{align}
\lvert z_2\rvert -\lvert z_1\rvert + \lvert z_1\rvert -\lvert z_2\rvert  &\leq \lvert z_2-z_1\rvert +\lvert z_1-z_2\rvert \\
0&\leq \lvert z_2-z_1\rvert +\lvert z_1-z_2\rvert \\
0&\leq 2\lvert z_1-z_2\rvert
\end{align}
Or if I subtract them:
\begin{align}
\lvert z_2\rvert -\lvert z_1\rvert -(\lvert z_1\rvert -\lvert z_2\rvert)&\leq \lvert z_2-z_1\rvert -( \lvert z_1-z_2\rvert )\\
2\lvert z_2\rvert-2\lvert z_1\rvert &\leq
\lvert z_2-z_1\rvert -\lvert z_1-z_2\rvert\\
2\lvert z_2\rvert-2\lvert z_1\rvert &\leq 0
\end{align}
I'm stuck here!

Hint:
$$|z_1-z_2| = |z_2-z_1|$$
