# Covering measure zero set by finitely many intervals which sum to desired length

I know that by definition of Lebesgue measure, if the set N has Lebesgue measure zero i.e. m(N) = 0 then we can cover the set by countably many open intervals {a$$_{j}$$, b$$_{j}$$}$$_{j\in\mathbb{N}}$$ such that $$\sum_{j}$$ b$$_{j}$$ - a$$_{j}$$ $$<$$ $$\epsilon$$.

I was wondering if the set N is contained in some bounded interval [a,b], would I be able to replace the countable open cover by finitely many open (left open or right open) intervals.

In other words, is it possible to say that I can cover N by finitely many intervals {a$$_{j}$$, b$$_{j}$$}$$_{j=1}^{n}$$ such that $$\sum_{j=1}^{n}$$ b$$_{j}$$ - a$$_{j}$$ $$<$$ $$\epsilon$$.

Here is an interesting exercise: see if you can prove that if $$N$$ is the set of rational numbers in between $$0$$ and $$1$$ and $$N \subset \bigcup_{k=1}^n (a_k,b_k)$$, then $$\sum_{k=1}^n (b_k - a_k) \ge 1$$.