Meromorphic functions on the extended plane

Recall that a divisor $D_f$ of a rational function $f(z)$ on $\mathbb {C}$ is a set $\left\{ p \in \mathbb {C} \cup \left\{ \infty \right\} \right\}$, consisting of zeros and poles $p$ of $f(z)$ (including the point $\infty$), counted with their multiplicities $n_p \in \mathbb {Z}$. Let $f$ and $g$ be two rational functions with disjoint divisors. Prove that

Prove that every meromorphic function on $S^{2}$ is rational.

Let $f(z)$ be a function, regular on the set $\overline{\mathbb {C}}$, except for a single simple pole $z = a$. Prove that if $a = \infty$, then $f(z)$ is a linear function, i.e., $f(z) = A z+B$, where $A \neq 0$, and if $a \neq \infty$, then $f(z)$ is a fractional-linear function, i.e., $f(z) = \frac{a z+b}{c z+d}$, where $a d-b c \neq 0$, $c \neq 0$.

Let $P_{m}(z)$ and $Q_{n}(z)$ be polynomials of degree $m$ and $n$, respectively, having no common zeros. Prove that the zeros of the polynomial $Q_{n}(z)$, and only these points, are poles of the function $f(z) = \frac{P_{m}(z)}{Q_{n}(z)}$, and the function $f(z)$ has no other singular points in the finite plane. Show that the point $z = \infty$ is a pole of order $m-n$ of the function $f(z)$ if $m > n$, and a point of regularity if $m \leqslant n$.

Let a meromorphic function $f(z)$ have, in $\overline{\mathbb {C}}$, only a finite number of poles $a_{1}, a_{2}, \ldots , a_{m}$ (the point $z = \infty$ may also be a pole). Prove that $f(z)$ is a rational function (a ratio of two polynomials) and is represented in the form

where $f_{k}(z)$ and $f_{0}(z)$ are, respectively, the principal parts of the Laurent series of the function $f(z)$ in the neighborhoods of the points $a_{k}$ and $z = \infty , A = \lim_{z \rightarrow \infty }\left(f(z)-f_{0}(z)\right)$.

Let an entire function $f(z)$ satisfy, in some domain $\left|z\right| > R$, the condition $\left|f(z)\right| \geqslant A\left|z\right|^{m}$, where $A > 0$ and $m \geqslant 0$ is an integer. Prove that $f(z)$ is a polynomial.