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, …