Every rational number can be written in one way

[topengonzo]

Homework Statement

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!

Homework Equations

None

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 doesn't 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 .

 

[HallsofIvy]

Looks to me like you can just construct it. For any rational number, x, let $a_1$ be the largest integer less than or equal to x. Then $x- a_1< 1$ so that $2!(x- a_1)= 2(x- a_1< 2$. Let $a_2$ be the largest integer less than or equal to $x- a_1$, etc.