Incompleteness of Cantor's enumeration of the rational numbers

Apply Cantor's enumeration of the rational numbers q_n, n = 1, 2, 3, ... Cover each q_n by the interval
ε[q_n - sqrt(2)/2^n, q_n + sqrt(2)/2^n].
Let ε --> 0.
Then all intervals together have a measure m < 2ε*sqrt(2) --> 0.
By construction there are no rational numbers outside of the intervals. Further there are never two irrational numbers without a rational number between them. This however would be the case if an irrational number existed between two intervals with irrational ends. (Even the existence of neighbouring intervals is problematic.) Therefore there is nothing between the intervals, and the complete real axis has measure 0.
This result is wrong but implied by the premise that Cantor's enumeration is complete.
Regards, WM