A proof that 0.999... = 1

The proof that 0.999... = 1 is elementary, using just the mathematical tools of comparison and addition of (finite) decimal numbers.

Problem

Write and explain a proof that 0.999... = 1. Use an elementary proof of the equation 0.999... = 1, with just the mathematical tools of comparison and addition of (finite) decimal numbers, without any reference to more advanced topics such as series, limits, formal construction of real numbers.

Solution

by AskAI 
The proof is by contradiction. Assume that 0.999... ≠ 1. Then there must be some number k between 0 and 1 such that 0.999... < k < 1. This implies that 9 < 10k < 10, which in turn implies that k = 1, a contradiction. Therefore, 0.999... = 1.
A.I. Evaluation of the Solution

This solution is correct and demonstrates a level of completeness. The approach is straightforward and uses only the mathematical tools of comparison and addition of decimal numbers.

Evaluated at: 2022-11-11 13:45:21