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

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

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

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

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

Or that the function h defined as:

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

h(z):= \lim_{z\to z_k}g(z) for 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?