If you want to be formal, 0.999... is by definition the limit as n goes to infinity of the sum from x=1..n of (9/10^x). This limit converges to 1. More intuiti

If you want to be formal, 0.999... is by definition the limit as n goes to infinity of the sum from x=1..n of (9/10^x). This limit converges to 1. More intuitively, what is the difference (as in subtraction) between 0.999... and 1? I think you'll find that it doesn't make sense for it to be anything other than zero. And if the difference between two numbers is zero then surely they must be equal.