Prove that for every $n \in \Bbb Q^{+}$, $\exists ! k \in \Bbb Z^{+}$ and a sequence of non-negative integers $\{a_m\}$ such that $a_1 \geq 0, a_k > 0$ and $0 \leq a_j < j, \forall j: \, 1 < j \leq k$ so that $n = \sum_{j = 1}^k \frac{a_j}{j!}$
Proving that it is unique is not a real problem, but proving that a representation exists is hard. My approach is to show that the sequence $b_1 = n$ and $b_j = b_{j-1} - \frac{\lfloor b_{j-1}(j-1)! \rfloor}{(j-1)!}$ is such that $b_k = 0$ for some k; then I’ll show it’s unique: but I couldn’t advance. How’d I prove it?