I'm reading the proof that a meromorphic function on the Riemann sphere must be a rational function, and I think I need to understand Laurent series better.
The idea of the proof I'm reading is that if $f$ is meromorphic on $\tilde{\mathbb{C}}$, then it has finitely many poles and we can subtract off the principal part of the Laurent series $\varphi_{i}$ at the poles $z_i$, then $f - \varphi_1 - \dotsb - \varphi_n -\varphi_\infty$ is analytic and bounded on $\varphi$.
How can we prove that if $\varphi_1$ is the principal part of the Laurent series of $f$ near the pole $z_1$, then $f-\varphi_1$ is analytic in a neighborhood of $z_1$? I think this should follow closely from the definition of the Laurent series.
A reference is sufficient, but I'm looking at the relevant sections of Ahlfors and Brown/Churchill and I feel like I'm getting lost in the epsilons.
If $f$ has a pole at $z_{0}$, then, for some $R > 0$, one has $$ f(z) = \sum_{n=-K}^{\infty}a_{n}(z-z_{0})^{n}=\frac{a_{-K}}{(z-z_{0})^{K}}+\cdots\frac{a_{-1}}{(z-z_{0})}+\sum_{n=0}^{\infty}a_{n}(z-z_{0})^{n},\;\;\; 0 < |z-z_{0}| < R. $$ Therefore $$ g(z)=f(z) -\frac{a_{-K}}{(z-z_{0})^{K}}-\cdots-\frac{a_{-1}}{(z-z_{0})}=\sum_{n=0}^{\infty}a_{n}(z-z_{0})^{n},\;\;\; 0 < |z-z_{0}| < R, $$ has a power series expansion near $z=z_{0}$ and, thus, a removable singularity at $z_{0}$. There are only a finite number of singular terms and, so, convergence is not an issue. The singular terms at $\infty$ are positive powers of $z$. Once you subtract all singular terms from $f$, including those at $\infty$ (a polynomial), then you end up with a function which has an entire, bounded extension and, as such, must be a constant. Absorbing any non-zero constant into the polynomial gives the partial fraction expansion $$ f(z) = p_{0}+p_{1}z+\cdots+p_{M}z^{M}+\sum_{l=1}^{L}\sum_{k=0}^{K_{l}}\frac{a_{l,k}}{(z-z_{l})^{k}}. $$