So I have to show $\mathbb{P}^1 \simeq \mathbb{C}_\infty$ using their charts.
Let $F: \mathbb{P}^1 \rightarrow \mathbb{C}_\infty$
be defined via $[z:w]$ $\mapsto$ $(\frac{\mathrm{Re}z \overline{w}}{\vert z \vert^2 + \vert w \vert^2}, \frac{\mathrm{Im}z \overline{w}}{\vert z \vert^2 + \vert w \vert^2}, \frac{\vert z \vert^2 - \vert w \vert^2}{\vert z \vert^2 + \vert w \vert^2})$
I showed it is well defined, and holomorphic already using complex charts $\varphi_0,\varphi_1$ on $\mathbb{P}^1$ and $\psi_0,\psi_1$ on $\mathbb{C}_\infty$ (won't get into defining these as it won't really matter for onto), they're the standard ones though for $\mathbb{P}^1$ they're $z/w$ and $w/z$ depending on which is nonzero, and $\psi$ are stenographic projection) .
I wanted to verify how I showed my map is onto:
For short let $\varphi,\psi$ be charts on $\mathbb{P}^1,\mathbb{C}_\infty$ resp.
Let $p \in \mathbb{C}_\infty$, we want to show $\exists q \in \mathbb{P}^1$ s.t. $F(q)=p$.
Consider $\psi(p) \in \mathbb{C}$ and note $\psi(p) \in$ Im $(\psi \circ F \circ \varphi^{-1})$,
That is, $\exists r \in \varphi^{-1}(U \cap F^{-1}(V))$ , (where $U,V$ are open sets of $\mathbb{P}^1, \mathbb{C}_\infty$ resp.)
Such that $(\psi \circ F \circ \varphi^{-1})(r) = \psi(p)$ Then we hit both left sides with $\psi^{-1}$:
$\psi^{-1} \circ (\psi \circ F \circ \varphi^{-1})(r) = \psi^{-1} \circ \psi(p)$ but $\circ$ is associative and $\psi$ is bijective thus
$(F \circ \varphi^{-1})(r) = p $ or
$F(\varphi^{-1}(r))=p$
Since $\varphi^{-1}(r) \in \mathbb{P}^1$, we can take $q =\varphi^{-1}(r)$ as we needed.
Does this work for onto?? And for $1-1$, I need to consider the points $[1,0],[0,1]$ correct?
Does all of this look ok?