# Meromorphic function $f$ with finite set of poles: existence of rational function $h$ s.t $p(h)=p(f)$?

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.

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.