Someone asked this question on another message board I go to. I’m asking it here because the other message board is mostly about video games and I figured someone here would have a better idea.

The first of your statements that I quoted is false, and can only be discussed using the principles of “The Calculus.” Your second quoted statement is true, following the basic rule of math that states that any number divided by itself equals unity, or 1.

While 1/3 can be said to be .333333… and 2/3 can be said to be .6666666…, it is silly to simply multiply the infinite decimally-expressed value of 1/3 by 3 to arrive at a statement like “3/3 = .9999~.”

Anyone who thinks it is untrue that 0.999… = 1 does not understand limits. (Which is not a terrible thing; lots of Calculus students don’t understand limits.)

Imagine you had a blackboard with 0.999 on it. And every second you added another 9 to the end. The number you would have at the “end of time” would not be the same as 0.999…

Also, it is quite possible for every entry in a sequence to be less than the limit of that sequence. That doesn’t mean that its limit is not its limit.

Q.E.D., please be a little more careful with your ellipses. They’re more important when discussing this problem than any other time, in my estimation.

Why is it silly?
Using “The Calculus” ( or analysis, if you prefer;specifically, properties about infinite series):
0.333… = [symbol]S[/symbol]3/10[sup]i[/sup] = 1/3 ( sum of an infinite G.P.)
and so
1 = 3/3 = 3*1/3 = 3[symbol]S[/symbol]3/10[sup]i[/sup] = [symbol]S[/symbol]9/10[sup]i[/sup] = 0.999…

It’s an infinite series. When they talk about it never reaching one, they are referring to an increasing, but still finite, number of 9s being tacked onto the end.

No, the paragraohs you quote are quite correct. It is an important feature of limits that they do noy have to reach their limiting value after any finite number of steps.

For a ( usually) less controversial example, consider the sequence 1/n. Its terms are
1, 1/2, 1/3, 1/4, 1/5, …
Most people would agree that this sequence tends to 0 as n tends to infinity, and they would be correct. The fact that no term of the sequence is actually 0 is wholly irrelevant.

Similarly, when we say that
0.999… = 1 we mean that 1 is the ( unique) limit of the sequence
0.9, 0.99, 0.999, 0.9999, …