$C(K,\mathbb{C})$ is the space of all continuous functions $f$ on a compact set $K$ of $\mathbb{C}$. Let $B(E)$ be the collection of $f$ in $C(K,\mathbb{C})$ such that there is sequence of rational functions $\{R_n(z)\}$ with poles only in $E$ such that $R_n \to f$ uniformly on $K$. Then $B(E)$ is a closed subalgebra of $C(K,\mathbb{C})$ that contains every rational function with poles in $E$.
It is easy to see that $B(E)$ is a subalgebra, i.e. if $f,g \in B(E),\,\, \alpha \in \mathbb{C}$, than $f+g,fg,\alpha f \in B(E)$
Prove $B(E)$ is closed in $C(K,\mathbb{C})$.
Attempt: Let $f \in \overline{B(E)}$. Then $f \in C(K,\mathbb{C})$ and there exists $f_n \in B(E)$ such that $f_n \to f$ uniformly on $K$. Now for each $f_n$, there exists a sequence of rational functions $R_{m,n}(z)$ with poles in $E$ such that $R_{m,n}\to f_n$ as $m\to \infty$ uniformly on $K$. How do I show that there is a subsequence of $R_{m,n}$ that converges to $f$ uniformly on $K$? There is no nesting of subsequences and so I can't use a diagonliaztion argument.
Any help will be appreciated.