Notation: with $\mathbb{C} ((z))((w))$ we denote the space of formal bilateral power series which have bounded below powers of $w$ but not uniformly bounded below powers of $z$ (this is in the context of vertex algebras, which I am reading about here, section 2.2.3). For example $$\sum_{n \leq 0} z^{n} w^{n} $$ doesn't belong to $\mathbb{C} ((z))((w))$, while $$\sum_{n \geq 0} z^n w^{n}$$ or $$\sum_{\substack{n \geq -m \\ m\geq -4}} z^n w^{m}$$ do belong to this space. Note that in the last example the power of $w$ is bounded from below, while the same is true for the power of $z$ only when $m$ is fixed and we consider the coefficient of $w^m$.
Now the point of the question: I'm trying to show that $\mathbb{C} ((z))((w))$ is a field, but this is not clear at all to me. I guess I have to formally expand something in order to find a somewhat explicit expression for $1/f$ where $f\in \mathbb{C} ((z))((w))$ is a generic element. Any help is greatly appreciated.