Quidditch wrote: Prove that any rational $r \in (0, 1)$ can be written uniquely in the form $$r=\frac{a_1}{1!}+\frac{a_2}{2!}+\frac{a_3}{3!}+\cdots+\frac{a_k}{k!}$$where $a_i\text{’s}$ are nonnegative integers with $a_i\leq i-1$ for all $i$. Wrong. $\frac{1}{2}=\frac{0}{1!}+\frac{1}{2!}=\frac{0}{1!}+\frac{1}{2!}+\frac{0}{3!}$.

If the condition "$a_k>0$" were added, it would be true.

Quidditch wrote: Prove that any rational $r \in (0, 1)$ can be written uniquely in the form $$r=\frac{a_1}{1!}+\frac{a_2}{2!}+\frac{a_3}{3!}+\cdots+\frac{a_k}{k!}$$where $a_i\text{’s}$ are nonnegative integers with $a_i\leq i-1$ for all $i$. it's here