Countable union of Jordan sets is not always Jordan measurable

[cragar]

Homework Statement

Show that the countable union or countable intersection of Jordan measurable sets need not be Jordan measurable, even when bounded.

The Attempt at a Solution

For countable intersection, I think the rationals from 0 to 1 will work, each rational have jordan measure zero.
But The jordan outer measure would be 1, because you would need to include the whole interval to contain all the rationals. For the countable intersection that seems more difficult. I am trying to think of a way to construct the vitali set using that. Well maybe for the intersection we take the unit interval [0,1] and shift it by that rationals of the form $\frac{1}{2^n}$

 

[Nino MMP]

to get sets {[0,1]-\frac{1}{2^n}}. Then the outer measure of the countable intersection would be zero, while the inner measure would be 1.