Q:

Prove that if $|z|<1$, then $|z^2-z+i|<3$

Prove that if $|z|<1$, then $|z^2-z+i|<3$. How can I prove this?

A:

Let $z\in\Bbb C$. Then $z^2\in\Bbb C$. Note that $|i|=1$, $|-z|=|z|$, and $|z^2|=|z|^2$. Thus, using the Triangle Inequality
$$\begin{align}|z^2-z+i|&=|z^2+(-z)+i|\\
&\leq |z^2|+|-z|+|i|\\
&=|z|^2+|z|+1\\
&<1^2+1+1\\
&=3.
\end{align}$$

