Points of a Measure Zero Sets Covered by Intervals Infinitely Many Times Given a measure zero set $E$, by definition we have forall $\varepsilon > 0$, a covering of $E$ by intervals whose lengths sum to $< \varepsilon$.
I want to prove that we can cover $E$ in intervals such that the sum of the lengths the intervals is $< 1$ and each point in $E$ is contained in infinitely many of the intervals.
Do you know how to prove this? Thank you
 A: Cover $E$ in intervals whose lengths sum to at most $\varepsilon$. Cover it again, but now with sum at most $\varepsilon/2$. Cover it again, but now with sum at most $\varepsilon/4$ … I think you begin to get the idea? Keep on doing this and take the union of all the covers, whose lengths sum to at most $\varepsilon$.
A: How about this:
First find a cover of $E$ which has sum of lengths less than 1/2,
The find another cover of $E$ which has sum of lengths less than 1/4, 
etc. etc.
The union of all the covers will then be a cover with sum of lengths less than 1, and we just need to show that each point of $E$ is in an infinite number of the covering intervals (it might happen that some of the intervals chosen at each step are the same as intervals chosen at a previous step).
To prove this last point, take a point $x$ and suppose it is contained in only finitely many (say $n$) of the intervals. Let the length of the smallest of these be $y$. Then there is an interval in a "later covering" (of total length less than $y$) containing  $x$, and it must be of length smaller than any of the $n$ intervals. This gives a contradiction.
