Order of the singularity of a sum of two functions with poles

Let \(f\) have a pole of order \(m\) at \(z_0\), and let \(g\) have a pole of order \(n\) at the same point. Show that \(f+g\) has either a removable singularity at \(z_0\) or a pole of order at most \[ \max\{m,n\}. \]