For part 1 we will consider just rationals between 0 and 1.

Fix some arbitrarily small epsilon.

Cover 1/2 with an interval of length eps/2, say (1/2 - eps/4, 1/2 + eps/4).

Cover 1/3 and 2/3 with two intervals, each of length eps/8.

Cover 1/4 and 3/4 with two intervals, each of length eps/16.

Cover 1/5, 2/5, 3/5, and 4/5 with 4 intervals, each of length eps/64.

For the different denominators, the sum of the lengths is e/2, e/4, e/8, e/16, ....