I have to show that the action $$ \lambda\cdot z:= \lambda z, $$ where $\lambda \in \mathbb{C}^*$, and $z\in \mathbb{C}^n-\{0\}$ is proper, i.e. the map $$ \mathbb{C}^* \times \mathbb{C}^n-\{0\} \to (\mathbb{C}^n-\{0\}) \times (\mathbb{C}^n-\{0\}), (\lambda,z)\mapsto (\lambda z,z) $$ is proper.
I tried showing that the map is open, which is sufficient as it is injective. I also tried to show that the preimage of a closed, bounded set is closed and bounded, but I couldn't come up with a proof.
Do you have any suggestions? Thanks in advance!