I would like to have the (simplest) proofs for the following theorem (there may be more than one interesting approach):

Let $f=\frac{p}{q}:\mathbb{C}\longrightarrow\mathbb{C}$ be a quotient of polynomials (of course, at some points it may be undefined) and $\bar{f}:\mathbb{S}^2\longrightarrow\mathbb{S}^2$ the natural extension to the Riemann sphere. Then $\text{deg}(\bar{f})=\max\{\text{deg}(p),\text{deg}(q)\}$.

Any (original) proof is welcome.

I would like to have a simple proof for the following result:

Let $f=\frac{p}{q}:\mathbb{C}\longrightarrow\mathbb{C}$ be a quotient of polynomials (of course, at some points it may be undefined). There is a natural extension to a map $\bar{f}:\mathbb{S}^2\longrightarrow\mathbb{S}^2$, considering $\mathbb{S}^2=\mathbb{C}\cup\infty$ the compactification of the complex numbers (by taking limits, perhaps some analysis is needed). Then $\text{deg}(\bar{f})$(the topological degree of maps between spheres)$=\max\{\text{deg}(p),\text{deg}(q)\}$.

Any idea is welcome.

I would like to have the (simplest) proofs for the following theorem (there may be more than one interesting approach):

Let $f=\frac{p}{q}:\mathbb{C}\longrightarrow\mathbb{C}$ be a quotient of polynomials (of course, at some points it may be undefined) and $\bar{f}:\mathbb{S}^2\longrightarrow\mathbb{S}^2$ the natural extension to the Riemann sphere. Then $deg(\bar{f})=deg(p)-deg(q)$.

Any (original) proof is welcome.

I would like to have the (simplest) proofs for the following theorem (there may be more than one interesting approach):

Let $f=\frac{p}{q}:\mathbb{C}\longrightarrow\mathbb{C}$ be a quotient of polynomials (of course, at some points it may be undefined) and $\bar{f}:\mathbb{S}^2\longrightarrow\mathbb{S}^2$ the natural extension to the Riemann sphere. Then $\text{deg}(\bar{f})=\max\{\text{deg}(p),\text{deg}(q)\}$.

Any (original) proof is welcome.