In the wikipedia page about Cousin's problems (https://en.wikipedia.org/wiki/Cousin_problems) is stated that "the case of one variable is the Mittag-Leffler theorem"; I'm a bit in trouble with this statement.
I'm able to show that, assuming that Cousin I problem can be solved in every (connected) open subset of $\mathbb{C}$, the Mittag-Leffler theorem follows, but I can't prove the other implication; it seems to me that, proving that the Cousin I problem in one variable can always be solved on an open domain, you don't use Mittag-Leffler theorem at all, but only the existence of partitions of unity and the fact that every open domain $\Omega \subset \mathbb{C}$ is a holomorphy domain (that is, there is a holomorphic function on $\Omega$ that cannot be extended outside $\Omega$). For a proof of this fact see for example Krantz "Function theory of several complex variables".
Assume the Mittag-Leffler theorem, and let $(f_i)$ be some Cousin data, $f_i$ meromorphic on $U_i\subset \mathbb{C}$; $f_i-f_j$ holomorphic on $U_i\cap U_j$; and $\displaystyle\bigcup_{i\in I}U_i = \mathbb{C}$.
Consider $E=\{a\in \mathbb{C} \mid \exists i\in I, a\in U_i \land f_i$ is not holomorphic at $a\}$. Then $E$ is discrete and closed.
Indeed take $a\in E$, $i$ as in the definition of $E$. $a$ is isolated in the set $\{z\in U_i\mid f_i$ is not holomorphic at $z\}$ because $f_i$ is meromorphic.
So let $U$ be an open set containing $a$, $U\subset U_i$ such that $f_i$ is holomorphic on $U\setminus\{a\}$.
Now assiume $z\in E\cap U$. Then for some $j, z\in U_j$ and $f_j$ is not holomorphic at $z$. Let $V= U\cap U_j$, then both $f_i$ and $f_j$ are defined on $V$ and $f_i-f_j$ is holomorphic, so $f_i$ isn't holomorphic at $z$; thus since $z\in U$, it follows that $z=a$.
Thus $E\cap U = \{a\}$, so $E$ is discrete.
Moreover, if $z\notin E$, then take some $i$ such that $z\in U_i$, and take an open set $U\subset U_i$ on which $f_i$ is holomorphic. Then clearly any $f_j$ is holomorphic on $U\cap U_j$, so that $U\subset \mathbb{C}\setminus E$ : $E$ is closed.
Finally, for $a\in E$ consider $i$ such that $a \in U_i$; and let $p_a(z)$ be the principal part of $f_i$ at $a$. By the hypothesis on $f_i-f_j$, it doesn't depend on the chosen $i$, as long as $a\in U_i$.
We can now apply Mittag-Leffler's theorem to get $f$ meromorphic on $\mathbb{C}$ such that $f-p_a$ has no singularity at $a$, for all $a\in E$; and such that the set of poles of $f$ is included in $E$.
Then $f$ is a solution for the Cousin problem, because $f-f_i = (f-p_a)-(f_i-p_a)$, which is holomorphic at $a$ as a difference of holomorphic functions; so $f-f_i$ is holomorphic at every point of $E\cap U_i$, and since both are holomorphic on $U_i\setminus E$, $f-f_i$ is holomorphic on $U_i$.