Example

Suppose we are given in $C$: (i) a sequence $(a_n)$ of distinct points such that the sequence $(|a_n|)$ is increasing and tends to $+\infty$; (ii) for each $n$, a rational function $z \to R_n(z)$ defined in $C - \{a_n\}$ and having a pole at $a_n$; (iii) an increasing sequence $(B_n)$ of open discs centred at 0, whose union is $C$, and such that none of the $a_k$ is on the frontier of any of the discs $B_n$. For each $n$, let $B'_n$ denote the intersection of $\overline{B}_n$ and the complement in $C$ of the set of points $a_n$; and let $X_n$ denote the set of all mappings

of $B'_n$ into $C$, where $P$ is the restriction to $B'_n$ of a function which is continuous in $\overline{B}_n$ and holomorphic in $B_n$. We define a metric in $X_n$ by putting

It is easily verified that $X_n$ is complete with respect to this metric. Finally, for $n \leq m$, we define a mapping $f_{nm} : X_m \to X_n$ such that if $S \in X_m$ then $f_{nm}(S)$ is the restriction of $S$ to $B'_n$. It is clear that the $f_{nm}$ are uniformly continuous and that $(X_n, f_{nm})$ is an inverse system of uniform spaces. This being so, an element of the inverse limit $X = \lim_{\leftarrow} X_n$ can be canonically identified with a meromorphic function $F$ in $\mathbf{C}$, whose only poles are the points $a_n$, and which is such that for each $n$, $F(z) - R_n(z)$ is holomorphic at $a_n$. The classical theorem of Mittag-Leffler asserts that $X$ is not empty; by virtue of Theorem 1, we have only to verify the condition (ML$_{nn}$) for all $n$. Let

be an element of $X_n$, where $P_n$ is continuous in $\overline{B}_n$ and holomorphic in $B_n$; for any $m \geq n$, let $Q_{mn}$ be the restriction of this latter sum is a holomorphic function in some neighbourhood of $\overline{B}_n$, hence (by Taylor's theorem) for each $\varepsilon > 0$ there is a polynomial $P_{mn}$ such that $|Q_{mn}(z) - P_{mn}(z)| \leq \varepsilon$ in $B_n$; if $S_m$ is the restriction of $S_n + Q_{mn} - P_{mn}$ to $B'_m$, we have $S_m \in X_m$ and $|S_m(z) - S_n(z)| \leq \varepsilon$ in $B'_n$. This completes the proof. \*