existence of the unique holomorphic function on removable singularity

Let $$z_0\in\mathbb{C}$$ and $$R>0$$. Suppose $$f(z):D_r(z_0) − \{z_0\}$$ is holomorphic and that there exists $$C>0$$ and some $$\epsilon> 0$$ such that $$|f(z)| \leq C|z − z_0| ^{−1+\epsilon}$$ for all $$z$$ near $$z_0$$. Show that the singularity of $$f$$ at $$z_0$$ is removable. That is, there exists a unique holomorphic function $$F:D_r(z_0)\to\mathbb{C}$$ such that $$F(z)=f(z)$$ for $$z\in D_r(z_0) − \{z_0\}$$.

I am not seeking for solutions. I want to understand the existence of the unique holomorphic function. Can someone help me out?

• Hint: Try to think in terms of the Laurent series expression of $f$ around $z_0$ then Jun 8 '20 at 15:27
THEOREM 1.2[Conway p.103] If $$f$$ has an isolated singularity at the point $$z = a$$, then $$a$$ is a removable singularity if and only if $$\lim_{z \to a}(z-a)f(z) = 0$$