0.999.../Proof by equivalence of Cauchy sequences

Assumptions

 * Construction of the real numbers from Cauchy sequences
 * Definition from Cauchy sequences
 * The limit of a geometric sequence

Proof
In this formalism the task is to show that the sequence of rational numbers
 * $$\left(1 - 0, 1 - {9 \over 10}, 1 - {99 \over 100}, \dots\right)

= \left(1, {1 \over 10}, {1 \over 100}, \dots \right)$$

has the limit 0.