Prove that every positive rational number can be expressed uniquely as a finite sum of the form
$$a _ { 1 } + \frac { a _ { 2 } } { 2 ! } + \frac { a _ { 3 } } { 3 ! } + \cdots + \frac { a _ { n } } { n ! } ,$$
where $a _ { n }$ are integers such that $0 \leq a _ { n } \leq n - 1$ for all $n > 1$.