A very informal proof: Let X = 0.999… Then 10X = 9.999… 10X - X = 9.999… - 0.999… = 9 = 9X Therefore X = 1. Being able to do the subtraction infinitely is a

A very informal proof: Let X = 0.999… Then 10X = 9.999… 10X - X = 9.999… - 0.999… = 9 = 9X Therefore X = 1. Being able to do the subtraction infinitely is a little hand-wavy, but should suffice as a demo for middle and high schoolers. More formal proofs involve limits that one wouldn't normally encounter until a course in real analysis. Wikipedia has a whole page devoted to the topic: http://en.wikipedia.org/wiki/0.999.. http://en.wikipedia.org/wiki/0.999...