HMMT 2011 S8

Let $n$ be a positive integer, and let $a_{1}, a_{2}, \ldots, a_{n}$ be a set of positive integers such that $a_{1}=2$ and $a_{m}=\varphi\left(a_{m+1}\right)$ for all $1 \leq m \leq n-1$. Prove that $a_{n} \geq 2^{n-1}$.