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Show that if $f$ is meromorphic in $D$ and has a finite set of poles, then there is a rational function $h$ with $p(h)=p(f)$ and $(f-h){\vert_D}\in\mathcal{O}(D)$

This is taken from Remmert's Theory of complex functions, page 320. I would like to be advised on how to prove it.

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If $\alpha$ is a pole of $f$ of order $m_\alpha$, then near $\alpha$, we can write $f(z)=\sum _{n=-m_\alpha}^{\infty}c_n(z-\alpha)^n=h_\alpha(z)+f_\alpha(z)$, with $f_\alpha(z)$ holomorphic and $h_\alpha(z)=\sum _{n=-m_\alpha}^{-1}c_n(z-\alpha)^n$.

Choose $$h(z)=\sum_{\alpha\in P}h_\alpha(z),$$ where $P$ denotes the set of poles of $f$, note by construction that $f(z)-h(z)$ is holomorphic around any $\alpha\in P$ and then $f(z)-h(z)$ is holomorphic everywhere.

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