Shrinking the diameter of Vitali Set

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 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 in such that 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)

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