Composition with polynomial/ same type of singularity

Question

Let $f\in O(D_1(0){}-\{0\})$ and $ p $ a non constant polynomial.

Then $f$ and $p(f)$ have the same type of singularity at $z_o=0 $.

I think its fairtly easy to Show that if $f$ has a singularity of type $A$ at $z_0$ then $p(f)$ has the same type of singularity.

Howerver, how would one proof that if $p(f)$ has a singularity of Type $A$ then $f$ has a singularity of type $A$ at $z_o=0$

Would appreciate a hint in the right direction.

Answer 1

Hint $$\lim_{z \to z_0} f(z) = c \Longleftrightarrow \lim_{z \to z_0} p(f(z)) = p(c)$$

$$\lim_{z \to z_0} f(z) = \infty \Longleftrightarrow \lim_{z \to z_0} p(f(z)) = \infty$$ then use the caracterisation of singularities thanks to limits.