another way of proving that 0.999... = 1 is by accepting the simple fact that if two numbers are different, there must be at least one number between them. for example, a number between 1 and 2 is 1.5, and a number between 0.9 and 1 is 0.95. since 0.999... has an infinite number of 9s, there cannot be another number after the "last" 9, meaning there is no number between 0.999... and 1. therefore, they are equal.