Let $$$a_1, a_2, \ldots, a_n$$$ be the array of $$$n$$$ positive numbers. Then
$$$ \sum_{p \in S_n} \frac{1}{a_{p_1}} \cdot \frac{1}{a_{p_1} + a_{p_2}} \cdot \frac{1}{a_{p_1} + a_{p_2} + a_{p_3}} \cdot \ldots \cdot \frac{1}{a_{p_1} + a_{p_2} + a_{p_3} + \ldots + a_{p_n}} = \frac{1}{a_{1}} \cdot \frac{1}{a_{2}} \cdot \ldots \cdot \frac{1}{a_{n}} $$$,
where the sum is taken over all permutations of size $$$n$$$.
I can prove it by induction, but it doesn't feel satisfying. Is there a combinatorial approach (maybe if $$$a_i$$$ are integers)? Or some simple transformation that would make this identity trivial? Is this well-known?
