let's call a line with infinity points the unit line, with length 1. if a line has 2 length, it has 2x more points than the unit line so it has 2infinity points and so on. since this proportionality holds, we can say that the length of a line is the amount of points it has devided by infinity. the length of a line with infinity - 1 points would be (infinity - 1)/infinity = 1 - 1/infinity = 1 - 0.000..1 = 0.99999999... therefore if 0.9999... = 1 the unit line and the line with one less point must be the same length. but for that to be true, every point would need to have a length of 0. however, if every point had a length of 0 then every line would have a length of 0, since 0 * infinity = 0, which is obviously not true, anyone who has seen a line knows this is not true. so the only possibility is that 0.9999... =/= 1. QED