Covering rational numbers by countable open intervals
I have a question asking to show that rational numbers in the closed interval [0;1] can be covered by a countable family of open intervals of total length less or equaly to 1/2.
I have some understanding, but I don't know how to solve the issue.
For the question, I thought that as the initial interval is close we can always find a smaller interval (or a disk) that has its center at the boundary point of the interval. Thus, as the interval is closed, we cannot find infinitely smaller disk that are contained in the interval.
Would you guys help me to formalize my thoughts and could you give me some ideas?