### Problem #15 - cover me not

Pt En < change language This problem that I am posting here today was inspired by an awesome video by 3blue1brown . Problem statement: for a given $\epsilon > 0$, is there a way for you to cover all the rational numbers in the interval $[0, 1]$ with small intervals $I_k$, such that the sum of the lengths of the intervals $I_k$ is less than or equal to $\epsilon$? In other words (with almost no words), for what values of $\epsilon > 0$ is there a collection $\{I_k\}$ of intervals such that $$\left(\mathbb{Q}\cap [0,1]\right) \subseteq \left(\cup_k I_k \right) \wedge \sum_k |I_k| < \epsilon$$ Solution: such a family of intervals always exists, for any value of $\epsilon > 0$. We start by noticing that the rational numbers in the interval $[0, 1]$ are countably many, which means I can order them as $q_1, q_2, q_3, \cdots$. If you haven't solved the problem yet, take the hint I just gave you and try to solve it. After enumerating the rationals inside