[proofplan]
We first record the two structural facts about a polydisc $P$: it is a Stein space, and it is contractible. Steinness and Cartan's Theorem B give $H^1(P,\mathcal{O}_P)=0$, which kills the additive Čech cocycle attached to any Cousin I datum. Contractibility gives $H^2(P,\mathbb{Z}_P)=0$, and the exponential sequence then combines this with $H^1(P,\mathcal{O}_P)=0$ to give $H^1(P,\mathcal{O}_P^*)=0$. Finally, the multiplicative Čech cocycle attached to any Cousin II datum is therefore a coboundary, so the local meromorphic equations glue to a global meromorphic solution.
[/proofplan]
[step:Realize the polydisc as a Stein and contractible space]
Let $a=(a_1,\ldots,a_n)\in \mathbb{C}^n$ and let $r_1,\ldots,r_n\in (0,\infty]$ be radii such that
\begin{align*}
P=\{z=(z_1,\ldots,z_n)\in \mathbb{C}^n: |z_j-a_j|<r_j \text{ for every } 1\leq j\leq n \},
\end{align*}
where the condition $|z_j-a_j|<r_j$ imposes no restriction when $r_j=\infty$. By the Steinness of Polydiscs, this open set $P$ is Stein.
Define the homotopy
\begin{align*}
H: P\times [0,1] &\to P \\
(z,t) &\mapsto a+(1-t)(z-a).
\end{align*}
For each $z\in P$, each $t\in [0,1]$, and each $1\leq j\leq n$ with $r_j<\infty$, we have
\begin{align*}
|H_j(z,t)-a_j|=(1-t)|z_j-a_j|<r_j.
\end{align*}
Thus $H$ is well-defined, $H(z,0)=z$, and $H(z,1)=a$. Hence $P$ is contractible.
Since $P$ is an open subset of $\mathbb{R}^{2n}$, it is paracompact and locally contractible. By the comparison theorem for constant sheaf cohomology and singular cohomology and the homotopy invariance of singular cohomology, the constant sheaf $\mathbb{Z}_P$ satisfies
\begin{align*}
H^2(P,\mathbb{Z}_P)=0.
\end{align*}
[guided]
We need two different features of $P$. The additive Cousin problem is controlled by holomorphic sheaf cohomology, so we need Steinness. The multiplicative Cousin problem is controlled by the exponential sequence, so we also need the topological vanishing $H^2(P,\mathbb{Z}_P)=0$.
Choose a center $a=(a_1,\ldots,a_n)\in \mathbb{C}^n$ and radii $r_1,\ldots,r_n\in (0,\infty]$ such that
\begin{align*}
P=\{z=(z_1,\ldots,z_n)\in \mathbb{C}^n: |z_j-a_j|<r_j \text{ for every } 1\leq j\leq n \}.
\end{align*}
When $r_j=\infty$, the $j$th coordinate condition is void. By the Steinness of Polydiscs, this polydisc is a Stein space.
Now we verify contractibility directly. Define
\begin{align*}
H: P\times [0,1] &\to P \\
(z,t) &\mapsto a+(1-t)(z-a).
\end{align*}
This map contracts every point linearly to the center $a$. To check that the contraction stays inside $P$, fix $z\in P$, $t\in [0,1]$, and an index $j$ with $r_j<\infty$. Then
\begin{align*}
|H_j(z,t)-a_j|=(1-t)|z_j-a_j|<r_j,
\end{align*}
because $0\leq 1-t\leq 1$ and $|z_j-a_j|<r_j$. If $r_j=\infty$, there is no coordinate restriction to check. Therefore $H$ is a well-defined continuous map $P\times[0,1]\to P$, with $H(z,0)=z$ and $H(z,1)=a$. This proves that $P$ is contractible.
The cohomology group used in the exponential sequence is sheaf cohomology with coefficients in the constant sheaf $\mathbb{Z}_P$. Since $P$ is an open subset of $\mathbb{R}^{2n}$, it is paracompact and locally contractible. The comparison theorem for constant sheaf cohomology and singular cohomology identifies $H^2(P,\mathbb{Z}_P)$ with singular cohomology $H^2_{\mathrm{sing}}(P,\mathbb{Z})$. Because $P$ is contractible, the homotopy invariance of singular cohomology gives
\begin{align*}
H^2(P,\mathbb{Z}_P)=0.
\end{align*}
[/guided]
[/step]
[step:Apply Cartan's Theorem B to vanish $H^1(P,\mathcal{O}_P)$]
Let $\mathcal{O}_P$ denote the sheaf of holomorphic functions on $P$. Since $P$ is Stein and $\mathcal{O}_P$ is coherent by the Oka Coherence Theorem, Cartan's Theorem B applies to the coherent analytic sheaf $\mathcal{O}_P$ on $P$. Therefore
\begin{align*}
H^q(P,\mathcal{O}_P)=0 \qquad \text{for every } q\geq 1.
\end{align*}
In particular,
\begin{align*}
H^1(P,\mathcal{O}_P)=0.
\end{align*}
[guided]
Let $\mathcal{O}_P$ be the sheaf assigning to each open set $U\subseteq P$ the ring $\mathcal{O}_P(U)$ of holomorphic maps $U\to \mathbb{C}$. We want to prove that the first sheaf cohomology group of this sheaf vanishes.
We use Cartan's Theorem B. Its hypotheses are that the underlying complex space is Stein and that the sheaf is coherent. The first hypothesis holds because $P$ is Stein by the Steinness of Polydiscs. The second hypothesis holds because $\mathcal{O}_P$ is coherent by the Oka Coherence Theorem. Therefore Cartan's Theorem B gives
\begin{align*}
H^q(P,\mathcal{O}_P)=0 \qquad \text{for every } q\geq 1.
\end{align*}
Taking $q=1$ gives the vanishing needed for the Cousin I problem:
\begin{align*}
H^1(P,\mathcal{O}_P)=0.
\end{align*}
[/guided]
[/step]
[step:Kill the additive Cousin cocycle to solve Cousin I]
Let $\mathfrak{U}=(U_\alpha)_{\alpha\in A}$ be an open cover of $P$, where $A$ is an index set, and let $(m_\alpha)_{\alpha\in A}$ be a Cousin I datum on $\mathfrak{U}$. Thus each $m_\alpha$ is a meromorphic function on $U_\alpha$, and for every $\alpha,\beta\in A$ the difference $m_\alpha-m_\beta$ is holomorphic on $U_\alpha\cap U_\beta$.
For every $\alpha,\beta\in A$, define the holomorphic map
\begin{align*}
c_{\alpha\beta}: U_\alpha\cap U_\beta &\to \mathbb{C} \\
z &\mapsto m_\alpha(z)-m_\beta(z).
\end{align*}
On every triple intersection $U_\alpha\cap U_\beta\cap U_\gamma$,
\begin{align*}
c_{\alpha\beta}+c_{\beta\gamma}+c_{\gamma\alpha}
=(m_\alpha-m_\beta)+(m_\beta-m_\gamma)+(m_\gamma-m_\alpha)=0.
\end{align*}
Thus $(c_{\alpha\beta})$ is a Čech $1$-cocycle with values in $\mathcal{O}_P$.
Since $H^1(P,\mathcal{O}_P)=0$, this cocycle is a coboundary. Hence there exist holomorphic maps
\begin{align*}
h_\alpha: U_\alpha &\to \mathbb{C}
\end{align*}
such that, on $U_\alpha\cap U_\beta$,
\begin{align*}
c_{\alpha\beta}=h_\beta-h_\alpha.
\end{align*}
For each $\alpha\in A$, define the meromorphic section $M_\alpha$ on $U_\alpha$ by
\begin{align*}
M_\alpha=m_\alpha+h_\alpha.
\end{align*}
On $U_\alpha\cap U_\beta$,
\begin{align*}
M_\alpha-M_\beta
=(m_\alpha-m_\beta)+(h_\alpha-h_\beta)
=c_{\alpha\beta}-c_{\alpha\beta}
=0.
\end{align*}
By the sheaf gluing axiom for meromorphic functions, the compatible family $(M_\alpha)_{\alpha\in A}$ glues to a meromorphic function $M$ on $P$. Moreover, on each $U_\alpha$,
\begin{align*}
M-m_\alpha=h_\alpha\in \mathcal{O}_P(U_\alpha).
\end{align*}
Therefore $M$ solves the given Cousin I problem.
[guided]
Take an arbitrary Cousin I datum. This means we have an open cover $\mathfrak{U}=(U_\alpha)_{\alpha\in A}$ of $P$, indexed by a set $A$, and meromorphic functions $m_\alpha$ on the open sets $U_\alpha$ such that the local principal parts agree modulo holomorphic functions. Formally, for each pair $\alpha,\beta\in A$, the difference $m_\alpha-m_\beta$ is holomorphic on $U_\alpha\cap U_\beta$.
The obstruction to gluing the $m_\alpha$ directly is their difference on overlaps. Define
\begin{align*}
c_{\alpha\beta}: U_\alpha\cap U_\beta &\to \mathbb{C} \\
z &\mapsto m_\alpha(z)-m_\beta(z).
\end{align*}
This map is holomorphic by the defining compatibility condition of a Cousin I datum. On a triple intersection $U_\alpha\cap U_\beta\cap U_\gamma$, the three overlap differences add to zero:
\begin{align*}
c_{\alpha\beta}+c_{\beta\gamma}+c_{\gamma\alpha}
=(m_\alpha-m_\beta)+(m_\beta-m_\gamma)+(m_\gamma-m_\alpha)=0.
\end{align*}
Thus $(c_{\alpha\beta})$ is a Čech $1$-cocycle with values in the sheaf $\mathcal{O}_P$.
The vanishing $H^1(P,\mathcal{O}_P)=0$ says that every such additive holomorphic cocycle is a coboundary. Therefore there are holomorphic maps
\begin{align*}
h_\alpha: U_\alpha &\to \mathbb{C}
\end{align*}
such that
\begin{align*}
c_{\alpha\beta}=h_\beta-h_\alpha
\end{align*}
on each overlap $U_\alpha\cap U_\beta$. These functions are the corrections we add to the local meromorphic functions.
For each $\alpha\in A$, define
\begin{align*}
M_\alpha=m_\alpha+h_\alpha.
\end{align*}
Each $M_\alpha$ is meromorphic on $U_\alpha$ because it is the sum of a meromorphic function and a holomorphic function. On $U_\alpha\cap U_\beta$, we compute
\begin{align*}
M_\alpha-M_\beta
=(m_\alpha-m_\beta)+(h_\alpha-h_\beta)
=c_{\alpha\beta}-c_{\alpha\beta}
=0.
\end{align*}
So the corrected local meromorphic functions agree on overlaps. By the sheaf gluing axiom for meromorphic functions, they glue to a meromorphic function $M$ on all of $P$. Finally, on each $U_\alpha$,
\begin{align*}
M-m_\alpha=h_\alpha,
\end{align*}
and $h_\alpha$ is holomorphic. This is precisely the condition that $M$ solves the Cousin I datum.
[/guided]
[/step]
[step:Use the exponential sequence to vanish $H^1(P,\mathcal{O}_P^*)$]
Let $\mathcal{O}_P^*$ denote the sheaf of nowhere-vanishing holomorphic functions on $P$, and let $\mathbb{Z}_P$ denote the constant sheaf with value $\mathbb{Z}$. For every open set $U\subseteq P$, define the exponential morphism
\begin{align*}
\operatorname{Exp}_U:\mathcal{O}_P(U)&\to \mathcal{O}_P^*(U)\\
f&\mapsto \left(z\mapsto e^{2\pi i f(z)}\right).
\end{align*}
Together with the inclusion of locally constant integer-valued functions into holomorphic functions, these morphisms give the exponential sequence
\begin{align*}
0\longrightarrow \mathbb{Z}_P\longrightarrow \mathcal{O}_P\stackrel{\operatorname{Exp}}{\longrightarrow}\mathcal{O}_P^*\longrightarrow 1.
\end{align*}
The associated long exact cohomology sequence contains
\begin{align*}
H^1(P,\mathcal{O}_P)\longrightarrow H^1(P,\mathcal{O}_P^*)\stackrel{\delta}{\longrightarrow}H^2(P,\mathbb{Z}_P).
\end{align*}
From the previous steps,
\begin{align*}
H^1(P,\mathcal{O}_P)=0
\qquad\text{and}\qquad
H^2(P,\mathbb{Z}_P)=0.
\end{align*}
Exactness at $H^1(P,\mathcal{O}_P^*)$ therefore gives
\begin{align*}
H^1(P,\mathcal{O}_P^*)=0.
\end{align*}
[guided]
The Cousin II problem is multiplicative, so the relevant sheaf is not $\mathcal{O}_P$ but $\mathcal{O}_P^*$. Let $\mathcal{O}_P^*$ be the sheaf assigning to each open set $U\subseteq P$ the group of nowhere-vanishing holomorphic maps $U\to \mathbb{C}^*$, and let $\mathbb{Z}_P$ be the constant sheaf with value $\mathbb{Z}$.
For each open set $U\subseteq P$, define
\begin{align*}
\operatorname{Exp}_U:\mathcal{O}_P(U)&\to \mathcal{O}_P^*(U)\\
f&\mapsto \left(z\mapsto e^{2\pi i f(z)}\right).
\end{align*}
The kernel consists exactly of locally constant integer-valued holomorphic functions, so together with the inclusion $\mathbb{Z}_P\to\mathcal{O}_P$ this gives the exponential sequence
\begin{align*}
0\longrightarrow \mathbb{Z}_P\longrightarrow \mathcal{O}_P\stackrel{\operatorname{Exp}}{\longrightarrow}\mathcal{O}_P^*\longrightarrow 1.
\end{align*}
Passing to sheaf cohomology gives a long exact sequence. The part we need is
\begin{align*}
H^1(P,\mathcal{O}_P)\longrightarrow H^1(P,\mathcal{O}_P^*)\stackrel{\delta}{\longrightarrow}H^2(P,\mathbb{Z}_P).
\end{align*}
We have already proved the two endpoint vanishings:
\begin{align*}
H^1(P,\mathcal{O}_P)=0
\qquad\text{and}\qquad
H^2(P,\mathbb{Z}_P)=0.
\end{align*}
By exactness, the kernel of the connecting map $\delta$ is the image of $H^1(P,\mathcal{O}_P)$, which is zero. But the target $H^2(P,\mathbb{Z}_P)$ is also zero, so every element of $H^1(P,\mathcal{O}_P^*)$ lies in the kernel of $\delta$. Hence every element of $H^1(P,\mathcal{O}_P^*)$ is zero:
\begin{align*}
H^1(P,\mathcal{O}_P^*)=0.
\end{align*}
[/guided]
[/step]
[step:Kill the multiplicative Cousin cocycle to solve Cousin II]
Let $\mathfrak{V}=(V_\lambda)_{\lambda\in \Lambda}$ be an open cover of $P$, where $\Lambda$ is an index set, and let $(f_\lambda)_{\lambda\in\Lambda}$ be a Cousin II datum on $\mathfrak{V}$. Thus each $f_\lambda$ is a nonzero meromorphic function on $V_\lambda$, and for every $\lambda,\mu\in \Lambda$ the quotient $f_\lambda/f_\mu$ is a nowhere-vanishing holomorphic function on $V_\lambda\cap V_\mu$.
For every $\lambda,\mu\in\Lambda$, define the holomorphic map
\begin{align*}
b_{\lambda\mu}:V_\lambda\cap V_\mu&\to \mathbb{C}^*\\
z&\mapsto \frac{f_\lambda(z)}{f_\mu(z)}.
\end{align*}
On every triple intersection $V_\lambda\cap V_\mu\cap V_\nu$,
\begin{align*}
b_{\lambda\mu}b_{\mu\nu}
=\frac{f_\lambda}{f_\mu}\frac{f_\mu}{f_\nu}
=\frac{f_\lambda}{f_\nu}
=b_{\lambda\nu}.
\end{align*}
Thus $(b_{\lambda\mu})$ is a Čech $1$-cocycle with values in $\mathcal{O}_P^*$.
Since $H^1(P,\mathcal{O}_P^*)=0$, this cocycle is a coboundary. Hence there exist nowhere-vanishing holomorphic maps
\begin{align*}
g_\lambda:V_\lambda&\to \mathbb{C}^*
\end{align*}
such that, on $V_\lambda\cap V_\mu$,
\begin{align*}
b_{\lambda\mu}=\frac{g_\mu}{g_\lambda}.
\end{align*}
For each $\lambda\in\Lambda$, define the meromorphic section $F_\lambda$ on $V_\lambda$ by
\begin{align*}
F_\lambda=f_\lambda g_\lambda.
\end{align*}
On $V_\lambda\cap V_\mu$,
\begin{align*}
\frac{F_\lambda}{F_\mu}
=\frac{f_\lambda g_\lambda}{f_\mu g_\mu}
=b_{\lambda\mu}\frac{g_\lambda}{g_\mu}
=\frac{g_\mu}{g_\lambda}\frac{g_\lambda}{g_\mu}
=1.
\end{align*}
Thus $F_\lambda=F_\mu$ on overlaps. By the sheaf gluing axiom for meromorphic functions, the family $(F_\lambda)_{\lambda\in\Lambda}$ glues to a meromorphic function $F$ on $P$. Moreover, on each $V_\lambda$,
\begin{align*}
\frac{F}{f_\lambda}=g_\lambda\in \mathcal{O}_P^*(V_\lambda).
\end{align*}
Therefore $F$ solves the given Cousin II problem.
[guided]
Take an arbitrary Cousin II datum. This means there is an open cover $\mathfrak{V}=(V_\lambda)_{\lambda\in\Lambda}$ of $P$, indexed by a set $\Lambda$, and nonzero meromorphic functions $f_\lambda$ on $V_\lambda$ such that the quotients $f_\lambda/f_\mu$ are nowhere-vanishing holomorphic functions on overlaps.
The obstruction is now multiplicative. Define
\begin{align*}
b_{\lambda\mu}:V_\lambda\cap V_\mu&\to \mathbb{C}^*\\
z&\mapsto \frac{f_\lambda(z)}{f_\mu(z)}.
\end{align*}
This map is holomorphic and nowhere zero by the compatibility condition in the Cousin II datum. On a triple intersection $V_\lambda\cap V_\mu\cap V_\nu$, multiplication of quotients gives
\begin{align*}
b_{\lambda\mu}b_{\mu\nu}
=\frac{f_\lambda}{f_\mu}\frac{f_\mu}{f_\nu}
=\frac{f_\lambda}{f_\nu}
=b_{\lambda\nu}.
\end{align*}
So $(b_{\lambda\mu})$ is a Čech $1$-cocycle with values in $\mathcal{O}_P^*$.
The vanishing $H^1(P,\mathcal{O}_P^*)=0$ says that every multiplicative nowhere-vanishing holomorphic cocycle is a coboundary. Therefore there are nowhere-vanishing holomorphic maps
\begin{align*}
g_\lambda:V_\lambda&\to \mathbb{C}^*
\end{align*}
such that
\begin{align*}
b_{\lambda\mu}=\frac{g_\mu}{g_\lambda}
\end{align*}
on each overlap $V_\lambda\cap V_\mu$. These $g_\lambda$ are multiplicative correction factors.
For each $\lambda\in\Lambda$, define
\begin{align*}
F_\lambda=f_\lambda g_\lambda.
\end{align*}
Each $F_\lambda$ is meromorphic on $V_\lambda$ because it is the product of a meromorphic function and a holomorphic function. On $V_\lambda\cap V_\mu$, compute
\begin{align*}
\frac{F_\lambda}{F_\mu}
=\frac{f_\lambda g_\lambda}{f_\mu g_\mu}
=b_{\lambda\mu}\frac{g_\lambda}{g_\mu}
=\frac{g_\mu}{g_\lambda}\frac{g_\lambda}{g_\mu}
=1.
\end{align*}
Thus $F_\lambda=F_\mu$ on all overlaps. By the sheaf gluing axiom for meromorphic functions, the compatible local meromorphic functions glue to a meromorphic function $F$ on $P$. Finally, on each $V_\lambda$,
\begin{align*}
\frac{F}{f_\lambda}=g_\lambda,
\end{align*}
and $g_\lambda$ is holomorphic and nowhere zero. This is precisely the condition that $F$ solves the Cousin II datum.
[/guided]
[/step]
[step:Specialize the result to $\mathbb{C}^n$]
The space $\mathbb{C}^n$ is the polydisc with center $0\in\mathbb{C}^n$ and radii $r_1=\cdots=r_n=\infty$. Applying the two conclusions already proved with $P=\mathbb{C}^n$ shows that every Cousin I problem and every Cousin II problem on $\mathbb{C}^n$ is solvable. This proves the theorem.
[/step]
