to (an, bn) but xn does not. The proof is complete since, in all cases, at least one real number in [a, b] has been found that is not contained in the given sequence. Cantor's proofs are constructive and have been used to write a computer program that generates the digits of a transcendental number. This program applies Cantor's construction to a sequence containing all the real algebraic numbers between 0 and 1. The article that discusses this program gives some of its output, which shows how the construction generates a transcendental. Example of Cantor's construction An example illustrates how Cantor's construction works. Consider the sequence: , , , , , , , , , ... This sequence is obtained by ordering the rational numbers in (0, 1) by increasing denominators, ordering those with the same denominator by increasing numerators, and omitting reducible fractions. The table below shows the first five steps of the construction. The table's first column contains the intervals (an, bn). The second column lists the terms visited during the search for the first two terms in (an, bn). These two terms are in red. Since the sequence contains all the rational numbers in (0, 1), the construction generates an irrational number, which turns out to be − 1. Cantor's 1879 uncountability proof Everywhere dense In 1879, Cantor published a new uncountability proof that modifies his 1874 proof. He first defines the topological notion of a point set P being "everywhere dense in an interval": If P lies partially or completely in the interval [α, β], then the remarkable case can happen that every interval [γ, δ] contained in [α, β], no matter how small, contains points of P. In such a case, we will say that P is everywhere dense in the interval [α, β]. In this discussion of Cantor's proof: a, b, c, d are used instead of α, β, γ, δ. Also, Cantor only uses his interval notation if the first endpoint is less than the second. For this discussion, this means that (a, b) implies a < b. Since the discussion of Cantor's 1874 proof was simplified by using open intervals rather than closed intervals, the same simplification is used here. This requires an equivalent definition of everywhere dense: A set P is everywhere dense in the interval [a, b] if and only if every open subinterval (c, d) of [a, b] contains