I have the Riemann sphere defined as following $(\mathbb{S}^2,[A]$) where A is an atlas defined by:
$A= \left\{(\mathbb{S}^2\backslash \{N\},\varphi),(\mathbb{S}^2\backslash \{S\},\psi)\right\}$
where $N=(0,0,1)$, $S=(0,0,-1)$ , $\varphi: \mathbb{S}^2\backslash \{N\} \rightarrow \mathbb{C}$ and $\psi: \mathbb{S}^2\backslash \{S\} \rightarrow \mathbb{C}$
$\hspace{65mm} (x,y,u)\rightarrow \frac{x+iy}{1-u} ,\hspace{15mm} (x,y,u)\rightarrow \frac{x-iy}{1+u}$
and I have to show that every holomorphic map $f:\mathbb{S}^2\rightarrow \mathbb{C}$ is constant.
I think that the idea of the exercise is to prove that for every $f$ the compositions $\varphi \space \circ f$ and $\psi \space \circ f$ are constant.
Can anyone help me or give me any hint? Thank you in advance.
Your idea is almost right. But instead, consider $f\circ\varphi^{-1}:\mathbb C\to \mathbb C$ and $f\circ\psi^{-1}:\mathbb C\to\mathbb C$. These are entire. Show that both are bounded. There are multiple ways to do this. You could show that the limits as $\vert z\vert\to\infty$ of both exists and is finite via a suitable change of coordinates, or you could show that $f$ itself is bounded by using topological features of the Riemannian sphere. Then use their boundedness to deduce that they are constant. Use this to show that $f$ itself is also constant.