Define a complex structure on the unit sphere $S^{2} \subset \mathbb{R}^{3}$ using stereographic projection charts $\varphi, \psi$. Let $U \subset \mathbb{C}$ be an open set. Show that a continuous non-constant map $F: U \rightarrow S^{2}$ is holomorphic if and only if $\varphi \circ F$ is a meromorphic function. Deduce that a non-constant rational function determines a holomorphic map $S^{2} \rightarrow S^{2}$. Define what is meant by a rational function taking the value $a \in \mathbb{C} \cup\{\infty\}$ with multiplicity $m$ at infinity.

Define the degree of a rational function. Show that any rational function $f$ satisfies $(\operatorname{deg} f)-1 \leqslant \operatorname{deg} f^{\prime} \leqslant 2 \operatorname{deg} f$ and give examples to show that the bounds are attained. Is it true that the product $f . g$ satisfies $\operatorname{deg}(f . g)=\operatorname{deg} f+\operatorname{deg} g$, for any non-constant rational functions $f$ and $g$ ? Justify your answer.