I'm currently reading a proof and the very first line starts "We begin by noting that, on using Cauchy-Schwarz and the fact that the number of ways of writing $m=m_1\cdots m_k$ is $O(m^\varepsilon)$ for any $\varepsilon>0$, we have ..." (here, $k\geq1$ is fixed, $m$ and the $m_i$ are positive integers)
Is there some intuition for why the number of ways of writing $m=m_1\cdots m_k$ is $O(m^\varepsilon)$? I can't find a source for this fact and would like to know why it is true. Any help would be appreciated.