0.999.../Proof by the geometric series formula

Assumptions

 * Definition from series
 * The geometric series formula

Proof
Using the series definition of the value of an infinite decimal,


 * $$0.999\ldots = 9\left(\tfrac{1}{10}\right) + 9\left({\tfrac{1}{10}}\right)^2 + 9\left({\tfrac{1}{10}}\right)^3 + \cdots.\,$$

This is a geometric series with a common ratio of 1/10. Applying the geometric series formula,


 * $$0.999\ldots = 9\left(\tfrac{1}{10}\right) + 9\left({\tfrac{1}{10}}\right)^2 + 9\left({\tfrac{1}{10}}\right)^3 + \cdots = \frac{9\left({\tfrac{1}{10}}\right)}{1-{\tfrac{1}{10}}} = 1.\,$$