Each bi of X differs from each number in the list at di. Using the diagonal method we have created a number X such that r1 < X < r2 and X ∉ L (Cantor’s proof).
1. Since X is an element of [r1, r2] we can rewrite [r1, r2] as [r1, … X, … r2].
2. We can now select X as the first cut point in the partitioning of [r1, r2] and form the relative bound of the conjoined interval pair [r1, [X], r2].
3. Once X is designated a relative bound it will be inserted into L per the algorithm and we’ll have
L = (r1, X, r2).
4. Having shown that X ∈ L, we can say that the original assertion, X ∉ L, leads to a contradiction and must be false.
5. Since the list of [r1, r2] created using the diagonal method was lacking X, and we have shown that by partitioning [r1, r2] using the interval sieve includes X as a member of the list of [r1, r2] we can assert that, at the limit, L will be complete and this ends the proof.