By using the following lemmas:
A countable union of countable sets is countable.
Cartesian products of integers are countable.
Wouldn't it be possible to prove the countability of the reals between 0 and 1, by partitioning them into countable sets each one uniquely identified by a Cartesian pair (m, n)?
Each set (m, n) contains all reals that have their first m digits sum equal to n and all their other digits zero.
By the second lemma above there is a countable such (m,n) that cover the interval [1,0]. And each such set is also countable since it contains a finite amount of numbers. Therefore by using the first lemma above wouldn't that prove that the the reals between 1 and 0 are countable?
Appreciate any feedback on the above on where I went wrong; I suspect it might be in how I used the second lemma to encode the above sets and show there is countable number of such sets.