HMMT 2011 T3

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 $1 \leq m \leq n-1$, where, for all positive integers $k, \varphi(k)$ denotes the number of positive integers less than or equal to $k$ that are relatively prime to $k$. Prove that $a_{n} \geq 2^{n-1}$.