If what you have a problem with is the statement that "0.9999... = 1", then there are two proofs of that: 1. Analytical 0.9999... = \sum_{n=1} ^{\infty} (9 *

If what you have a problem with is the statement that "0.9999... = 1", then there are two proofs of that: 1. Analytical 0.9999... = \sum_{n=1} ^{\infty} (9 * 0.1^n) = 0.9/(1 - 0.1) = 0.9/0.9 = 1 The first equality is by definition of strings of the form "0.abcdef...". The second is the sum of geometric series. 2. Algebraical x = 0.9999... 10x = 9.99999... 9x = 10x - x = 9.9999... - 0.9999... = 9 x = 1