I have this problem with two parts with one part easy and obvious, the other part so hard. The problem goes as follows.
Consider the set $A=[0,1]\cap \mathbb{Q}$ and Lebesgue measure $\mu$.
a) Prove that for all $\epsilon>0$ there exists a covering $(A_k)_{k\in\mathbb{N}}$ of open intervals for A, such that $\sum_k \mu(A_k)<\epsilon$.
b) If $(A_k)_{k\in\{1,2,...,N\}}$ is a finite open covering for A, then $\sum_k \mu(A_k)>1$.
My thoughts for a) are as follows: Let $(r_n)\subseteq \mathbb{R}$ be chosen later. Arrange the set $A=\{a_1,a_2,\ldots\}$
For $\epsilon>0$ we let $A_k=(a_k-r_k\epsilon,a_k+r_k\epsilon)$ and then choose $(r_n)$ such that $\sum_k 2r_k<\epsilon$. This I can do :)
For b) though... I am at a loss. Any suggestions?