Q:

Sequence of subsets $E_n$ of $[0, 1]$ with $m(E_n) = 1$ for every n but $m(\cap_{n=1}^\infty E_n) \neq 1$

This is actually a "prove or give counterexample" type of problem.  The claim is that if any sequence of subsets $E_n$ of $[0, 1]$ have $m(E_n) = 1$ (m being the standard Lebesgue measure) for every n, then $m(\cap_{n=1}^\infty E_n) = 1$, but I don't think this is true.  My instinct says that there should be some way to construct even just a fairly small, finite sequence of subsets of this kind whose intersection has measure $0$, but I'm having trouble actually doing so.  The sequence ${E_n}$ is not a descending sequence, so there's no reason I can think of why the limit of their measures should be the measure of the intersection limit. 
Is there an "obvious" example I'm missing, or is my reasoning for why this shouldn't be true faulty? 

A:

Hint: What do De Morgan's laws say about $[0,1] \setminus (\bigcap_n E_n)?$

