Suppose $f(z)$ has an isolated essential singularity at $z = 0$ and $f(z) \neq 1$ for $ 0 < |z| < r$. I'm trying to prove that $1/(f(z) − 1)$ has an essential singularity at $z = 0$.
Honestly I'm not pretty sure where should I start on this proof. Should I eliminate the other two types of singularity, or is there a relevant theorem I should consider using? Thanks:)