Let $\displaystyle U\subset \mathbb{C}$ open and let $\displaystyle f$ be holomorphic on $\displaystyle U\setminus\left\{z_1\cdots z_n\right\} $

Having poles $\displaystyle z_1,\cdots,z_n$ of order $\displaystyle m_i>0$ for $\displaystyle i=1,\cdots ,n $

I'd like to show $\displaystyle g(z):=(z-z_1)^{m_1}\cdots(z-z_n)^{m_n}f(z)$ has removable singularities $\displaystyle z_1,\cdots, z_n$

I'm not so sure exactly what needs to be shown here:

For example that $\displaystyle \lim_{z\to z_k}g(z)$ exists?

Or that the function h defined as:

$\displaystyle h(z):=g(z)$ for $\displaystyle z\in U\setminus\left\{z_1,\cdots, z_n\right\}$

$\displaystyle h(z):= \lim_{z\to z_k}g(z)$ for $\displaystyle z_k\in \left\{z_1,\cdots, z_n\right\}$

is holomorphic on U?

I'm not really sure....or do I need to show something about the powerseries of g?