$\mathbb C^{\infty}\setminus\{0\}$ retracts on $S^{\infty}$

Question

We want to show that $\mathbb C^{\infty}\setminus\{0\}$ retracts on $S^{\infty}$.

$C^{\infty}\setminus\{0\}$ should be the same as $\mathbb R^{\infty}\setminus\{0\}$. Maybe the retraction could be $(1-t)x + t \frac{x}{|x|}$?