Lemma: Meromorphic functions on the Riemann sphere are just the rational functions in $z$
Full proof of the Lemma can be find in p.2: http://math.uh.edu/~minru/6322-12/ruRSnote.pdf
And here is the part I don't understand.
If $z_1, \dots , z_n \in C$ are the poles of f of degrees $d_1,\dots,d_n$ then
$$p := f (z-z_1)^{d_1}(z-z_2)^{d_2}......(z-z_n)^{d_n}$$
has no poles on $\mathbb{C}$ and has at most a pole at $\infty$. This means that $p$ is a polynomial
What does a pole at infinity means and is this really important to conclude that $p$ is a polynomial? And I think it is since there are other functions that doesn't have poles and isn't a polynomial. For example $\sin(z)$.