First of all, just a quick note: the transitivity of equality (A = B and B = C -> A = C) DOES hold in the reals, but it does NOT in many other sets.
>>138
No, that's the WHOLE difference between the rationals and the reals. In the reals, "be infinitely close" is the same as "be the same". That follows from the completeness of the uniform structure.
>>135
Explain me exactly HOW do you intend to equate 0.999... to 0.999...8. in crude terms, you can't really put something "after an infinity of 9's". However, I'm willing to give you the chance: how do you prove that 0.999... = 0.999...8 ?
Really guys, I'm make it a little clearer now: in the reals, there is NO SUCH thing as the "last number before a number". Between any two different numbers, there is an infinity of other numbers. Think a little about the following related question:
Which is the smallest number greater than 0?
Remember that 0 = 0.000... Is any of the guys talking about "0.999...8" willing to say that the smallest number greater than 0.000... is 0.000...1?