Prove inequality for complex numbers

Question

Can anyone help me or at least give me some hints
how to prove following inequality :

$ \lvert z-a\rvert \le \lvert \lvert a\rvert-\lvert z\rvert\rvert + \lvert z\rvert*\lvert arg (z/a)\rvert $

for any $ z, a \in \Bbb C, a \neq 0$

Answer 1

I hope this image helps as a hint.

The messy algebraic part seems proving that $$\left|z-|a|\cdot e^{i\arg z}\right|\le|z|\cdot\left|\arg\left(\frac za\right)\right|$$ (That is, the arc is longer than the chord).

Answer 2

Hint:
By dividing $|a|$ from both sides you can assume $a=1$. What is the geometric meaning of that?