From $|z_2|\leq M_2$, $\left|z_1-z_2\right|\leq M+M_2$ what could we say about $|z_1|$?

Question

Let $z_1$, $z_2\in\mathbb C$ and $M$, $M_2\in\mathbb R$. If $|z_2|\leq M_2$ and $$\left|z_1-z_2\right|\leq M+M_2\, ,$$ what could we say about $|z_1|$? For example, is it less than or equal to $M$?

However, using Triangle Inequality applied to $z_1-z_2$ and $z_2$ I could write: $$|z_1|=\left|\left(z_1-z_2\right)+z_2\right|\leq \left|z_1-z_2\right| + |z_2| \leq M+2M_2.$$ Is it the only thing I can say about the bound of $|z_1|$?

Answer 1

Yes, thats the only thing. Take for example $z_2=M_2 \ge 0$ and $z_1=M+2M_2 \ge 0$.