The proof I like the most: Clearly 0.9999... <= 1. Assume 0.9999... != 1 (*). Then 0.9999... < 1, so there must be some positive number P so that 0.9999... + P = 1. But for ANY NONZERO positive P, 0.9999... + P > 1, which is a contradiction, and definitely wrong. Therefore we are forced to conclude that the assumption (*) was incorrect, which gives us: 0.9999... = 1 Try refuting this one... http://mathforum.org/dr.math/faq/faq.0.9999.html http://qntm.org/pointnine Discuss.