> There is no proof that will ever satisfy a person dead-set against this. Yes there is. There is a proof that uses only fundamentals of first year university

> There is no proof that will ever satisfy a person dead-set against this. Yes there is. There is a proof that uses only fundamentals of first year university analysis. When you see 0.99999.... this can be written as an infinite sum \sum_{i=0}^\infty 0.9 x 10^{-i}. Truncate the sum and set S_n = \sum_{i=0}^n 0.9 x 10^{-i} and now simply use the rules of arithmetic progressions to get the limit out: 0.1 S_n = \sum_{i=0}^n 0.9 x 10^{-i-1} S_n - 0.1 S_n = 0.9 - 0.9 x 10^{-n-1} 0.9 S_n = 0.9 ( 1 - 0.1^{n+1} ) S_n = 1 - 0.1^{n+1} Now let n tend to infinity to find the limit, which is 1. You don't need to imagine infinity to do any of this.