if $|z| \leq 1$, which of the following statements must be true ? Indicate all such values
A. $z^2 \leq 1$
B. $z^2 \leq z$
C. $z^3 \leq z$
The book states its answer A, considering the numbers $\frac{1}{2}$ and $-\frac{1}{2}$
But if $|z| \leq 1$, then $z \leq 1$ and $-z \leq 1$
If we take the last condition $-z\leq1$ , or $z \geq -1$
Then we have to consider all numbers greater than or equal $-1$, for example $2$, which does not satisfy answer A. What gives ?
$ z \geq - 1 $ is necessary but not sufficient. Both $ z \geq -1 $ and $ z \leq 1 $ must be true for $ \left | z \right | \leq 1 $ to hold (note: although you're using the variable $ z $, I'm assuming $ z $ is real, since you are comparing $ z $ to numbers like $ 1 $, which cannot be done for complex numbers.)