Let $C=\{(x,\:y,\:z)\in \mathbb{R}^3: x^2+y^2=1,\; -1<z<1\}$ and $C=\{(x,\:y,\:z)\in \mathbb{R}^3: x^2+y^2+z^2=1,\; -1<z<1\}$. Define $F:C\to S$ as $$F(x,\:y,\:z)=(x\sqrt{1-z^2},\:y\sqrt{1-z^2},\:z)$$
Prove that $F$ is a diffeomorphism.
My attempt: The inverse of $F$ is given as $F^{-1}(x,\:y,\:z)=(\dfrac{x}{\sqrt{1-z^2}},\:\dfrac{y}{\sqrt{1-z^2}},\:z)$.
Now it is clear that both $F$ and $F^{-1}$ are smooth, so $F$ is diffeomorphism.
However, I figured out that my solution is incorrect. Why is my solution wrong and what is the right solution?
Any advices and hints are welcome. Thank you for your help!