Claim: The diameter of Vitali Set V on [0, 1] can be shrinked as much as we please
Proof:
Let R be an equivalence relation defined on [0,1] such that x is related to y if x – y is rational.
Let E be an equivalence class corresponding to the equivalence relation R.
In order to create the Vitali Set V (on [0,1] ) we need to pick exactly one element from each such equivalence class.
We will show that for an arbitrary equivalence class E and for any natural number n, it is possible to find an element z in E such that z < 1/n
Suppose z’ is in E and 1/n < z’
By Archimedean property we can say that there exists a natural number p such that p/n > z’
Let P be the set of all numbers such that p/n > z’
Now by well ordering principle P has a least element p*.
Hence (p* -1)/n < z’ < p*/n
Now z’ – (p*-1)/n < p*/n – (p*-1)/n= 1/n
Now z’ – (p*-1)/n is in the equivalence class of z’ as z’ – (z’ – (p*-1)/n ) = (p*-1)/n which is rational.
So we have found a number in the equivalence class E which is less than 1/n some arbitrary natural number n.
There fore the diameter of the Vitali Set V can be made smaller than 1/n for arbitrary natural number n.
(Proved)
