its wrong.
by changing the first digit of the first number, second digit of the second number and constructing a number out of that, the resulting number is supposed to not be in the set because it is completely different from every other number in that set.
but since the already existing set is infinite there is absolutely no reason for that new number not to be in the set. it will be pretty much impossible to find of course, but since the set contains all binary numbers, that "new" number will be found somewhere in the set. and yes, that "new" number that is already in the set will again be changed, but that would just create a new "new" number that is also in the set.

if im wrong please somebody correct me, the fact that this argument is used in so many places but is seemingly so easily disproven is driving me crazy.