You don't get it do you? Let me put it in an informal way, its the numbers Decimal Representation that is infinite, not the number itself. The number is by definition finite. You are making such a mess in your head about this, mixing completely unrelated stuff.
Adrian's proof its a series, or a sum, not a sequence. 1 is not infinite, it has two decimal representations 1.000.... and 0.999..... both infinite, but the Real numbers themselves are the same and finite.
Tell me, you know what a dense order is? Its a property of the Real Set and states: for all a,b that belong to the set, where a < b, there exists a number c so that a < c < b. Make a = 0.999... and b = 1, now, if you're right about 0.999... =/= 1 surely you have the conditions met a < b, now, find me a number c so that 0.999... < c < 1.
Adrian's proof its a series, or a sum, not a sequence. 1 is not infinite, it has two decimal representations 1.000.... and 0.999..... both infinite, but the Real numbers themselves are the same and finite.
Tell me, you know what a dense order is? Its a property of the Real Set and states: for all a,b that belong to the set, where a < b, there exists a number c so that a < c < b. Make a = 0.999... and b = 1, now, if you're right about 0.999... =/= 1 surely you have the conditions met a < b, now, find me a number c so that 0.999... < c < 1.