Decomposition of positive rationals as a series (uniqueness part)

Question

This is this question but I'm having troubles with the uniqueness part (there's some explanation on how to prove the uniqueness part but honestly I don't understand it). Here's the claim and my attempt

Any positive rational number $r$ can be expressed in exaclty one way in the form $$ r = \sum_{j=1}^n a_j/j!$$ where $a_1,a_2,\dots,a_n$ are integers such that $a_1\geq0$, $0\leq a_j <j$ for $2\leq j \leq n$ and $a_n\neq 0$.

As this is an exercise from Stromberg I think the uniqueness part must have two prove two facts:

First: Let $r = \sum_{j=1}^n a_j/j! = \sum_{j=1}^m b_j/j!$ with $n<m$. At this point I compute

$$ 0 = (a_1 - b_1)/1! + (a_2 - b_2)/2! + \dots + (a_n - b_n)/n!+b_{n+1}/(n+1)!+\dots+b_m/m! \neq 0 $$

because at least $b_m\neq0$ by hypothesis and this contradiction yields $n=m$ (is this correct?).

Second: With $r = \sum_{j=1}^n a_j/j! = \sum_{j=1}^n b_j/j!$ assume $a_j\neq b_j$ for some $j$ and let $s$ be the largest such $j$: say $a_s<b_s$. Here again

$$ 0=\sum_{j=0}^s (a_j-b_j)/j!=\dots $$

After this I don't know how to continue and arise a contradiction...

Answer 1

Here is a much more general theorem which I proved over 50 years ago.

Let $(B_n)_{n=1}^{\infty} $ be an increasing sequence of positive integers with $B_1 > 1$.

Then every real $0 \le r < 1$ can be represented in the form $r =\sum_{n=1}^{\infty} d_n/B_n $ where the $d_n$ are integers such that $0 \le d_n \lt B_{n+1}/B_n $ and the representation is unique for all $r$ if and only if $B_n | B_{n+1}$ for all $n$.

"unique" means that if a number has two representations then one is terminating and the other replaces the last digit $d_n$ with $d_n-1, m_{m+1}, m_{n+2}, ... $ where $m_n$ is the max value $d_n$ can be, $m_n = \lceil B_{n+1}/B_n \rceil -1 $. An example is $0.1 = 0.09999....$.

This is why decimal, binary, and factorial representations are unique.

I'll look and see if I still have that writeup. Until then, consider this an exercise.