0.999.../Proof by nested intervals

Assumptions

 * Definition from nested intervals

Proof
Applying the nested-intervals definition of the value of an infinite decimal, 0.999... is the unique real number that lies in all of the intervals [0, 1], [0.9, 1], [0.99, 1], and [0.99...9, 1] for every finite string of 9s. Since 1 is an element of each of these intervals, 0.999... = 1.