For every r>0, there is a vitali set having outer measure less than r

Question

For every $r>0$, there is a vitali set having outer measure less than $r$.

Here is my approach:

Take arbitrary positive real number r.

The interval $(0,r)$ has positive outer measure and hence contains a vitali set $V$(say)

Outer measure of $V$ is less than or equal to $r$.

But I need to construct a vitali subset whose outer measure is strictly less than $r$.

Answer 1

Do exactly what you did, but with an interval of length $r/2$.