Let $f(z)=g(z)h(z)$ such that $g$ has an essential singularity in $z=z_0$ and $h$ is holomorphic in a neighbourhood of $z_0$ then $f$ has an essential singularity in $z=z_0$?

Im trying to see this without using Laurent series expansion. I was thinking about the following, $g$ has an essential singularity in $z_0$ if and only if

$$\lim_{z\to z_0}g(z)$$

doesn't exist. On the other hand, since $h$ is holomorphic in $z_0$, thus

$$\lim_{z\to z_0}h(z)=l\in \Bbb{C}$$

However Im not sure if this necessairily implies that

$$\lim_{z\to z_0}g(z)h(z)$$

I wasn't able to find a counter example nor to prove it. Is it true?


Assume that $g(z)h(z)$ has a pole of order $k\geq 0$ at $z_0$ and $h(z)$ has a zero of order $m\geq 0$. Then: $$(z-z_0)^{k+m}g(z)$$ is regular at $z_0$, which is a contradiction.

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