[cokernel]

(Edited to try to answer the objection more clearly.)

The intervals overlap heavily. Each of the intervals R_i contains infinitely many rational numbers, and all but finitely many of those will have corresponding intervals which are completely contained within the interval R_i. As a result, the estimate obtained by the construction described is always strictly smaller than the chosen e.

[JadeNB]

I appreciate your careful and patient explanation, and assay a superficial and snarky one of my own: it is impossible for the measure of the rationals, if it is to exist at all, to be exactly `e` for every small positive real number `e`. Even if we didn't know anything about the overlap of the intervals, we'd know at best that we had a (possibly non-strict) upper bound of `e`; but the only non-negative real number that can satisfy such a bound for every positive `e` is `0`.