(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove that every positive rational number x can be written in ONE way in form

x=a1+ a2/2! + a3/3! + ..... + ak/k!

where a1,a2,...,ak are integers and 0<=a1, 0<=a2<2,.... ,0<=ak<k

I wrote my solution below. Please check if it is correct and rewrite it for me in a neater way. Thank you!

2. Relevant equations

None

3. The attempt at a solution

I proved by induction a1 > a2/2! > a3/3! > a4/4!>.... ak>k! when taking a1,a2,a3,..., ak not= 0 ( I started with 2 as base case since a1 has different interval from them). Can I prove it without induction since I don't use the term before?

Thus I notice that every fraction has a unique interval for a1,a2,a3,...,ak not = 0

and when it is 0, the fraction is 0 and doesnt add up. Thus it is unique

I also need to prove that every rational number can be written in this form. I take x=p/q and I add up the fractions to get that q=k! and p=a1 k!+ a2 k!/2! + .... + ak .

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# Every rational number can be written in one way

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