Prove inequality for all complex numbers

Question

Prove following inequality for all complex numbers: $ \lvert z\rvert \le \lvert z \rvert ^2 + \lvert z-1 \rvert $

It is obvious for $\lvert z\rvert \gt 1 $ but what about the rest ?
Any hints would be appreciated.

Answer 1

Hint: Write the inequality as $|z| (1 - |z|) \le |z-1|$. If $|z| \le 1$, $|z| (1-|z|) \le 1 - |z|$.

Answer 2

Try the triangle inequality $$ |z| = |z-z^2+z^2| \le |z-z^2|+|z^2|=|z||z-1|+|z^2|\le|z-1|+|z^2| $$ when $|z|\le1$.

Note also, $|z^2|=|z|^2$.