Cantor's list contains real numbers r as binary or decimal fractions. Real numbers, however, are /limits/ of binary or decimal fractions.
For every terminating fraction of r, Cantor obtains a difference between r and the due terminating fraction of the anti-diagonal d: r_nn =/= d_n. He concludes that this remains true for the limits of the list numbers r and d by using the argument: different sequences have different limits. But it is well known that this argument is not admissible in proofs because it is false.