If $\left | z \right | \leq 1$, which of the following must be true?

Question

If $\left | z \right | \leq 1$, which of the following must be true?

Indicate all such statements

A. $z^{2}\leq1$

B. $z^{2}\leq z$

C. $z^{3}\leq z$

Not sure if this is right, but is $\left | z \right | = z^{2}$?

For A), I have $-1\leq z \leq 1$ from $\left | z \right | \leq 1$. But I get $z \leq \pm 1$ from $z^{2}\leq1$.

For B), I get $z \leq 0$ and $z\leq 1$ from $z^{2}\leq z$.

For C), I get $z \leq 0$ and $z \leq \pm 1$ from $z^{3}\leq z$.

I know that the answer is A, but I don't understand why.

Answer 1

$$\begin{eqnarray*} &|z|\le 1 \iff-1 \le z \le 1 \\ \\A.& \;(z+1)(z-1)\le 0 &\iff -1 \le z \le 1 \\B.& \;z(z-1)\le 0 &\iff 0 \le z \le 1 \\C.& \;z(z-1)(z+1)\le 0 &\iff z\le -1 \text{ or } 0 \le z \le 1 \end{eqnarray*}$$

Statement A. is the only one that is true whenever $|z|\le 1$