To the mathematician, the answer to the above question is a clear "yes"; as sure as the fact that the infinite decimal .3333... represents 1/3. However, so many people think that .9999... represents something "a tiny bit less than 1" that I started wondering why; and I think that the following is the reason.
When someone says, "I am 99.99 percent sure", or "We have .9999 of the information we need", these are correct ways of saying, "a tiny bit less than all", since .9999 denotes 9,999/10,000, i.e., 1 minus 1/10,000. The more 9's one adds, the closer one is to 1; each additional 9 cuts the difference from 1 by a factor of 10, but no finite number of 9's brings one exactly to 1. People hear such expressions frequently, and may not realize the subtle distinction between .9999, i.e., exactly four 9's, after the decimal point (or .99999, i.e., exactly five 9's, etc.) and ".9999...", where the three dots mean that the 9's are understood to go on forever. But by the mathematical interpretation of infinite decimals, the number represented by .9999... is the value which the infinite sequence of numbers .9, .99, .999, etc. is approaching; namely, 1.
Going back to the use of .9999 to mean "a little bit less than 1", the people who say this usually have not computed the discrepancy from 1 exactly, and found that it is 1/10,000. Rather, they use such expressions loosely, not worrying exactly how many 9's they say. Doubtless most people learn, from hearing how this expression is used, that "a decimal point followed by an uncertain number of 9's" means a very tiny bit less than 1, perhaps before they learn about infinite decimals. If one doesn't think about it, it is easy to confuse "a decimal point followed by an uncertain number of 9's" with "a decimal point followed by an endless string of 9's".
The same distinction exists between, say, .3333, which is a tiny bit less than 1/3, and .3333..., which is exactly 1/3. But people hardly ever have occasion to say, "I got almost but not quite a third of it", and express this by ".3333". If that were common usage, people might likewise wonder, "Is the infinite decimal .3333.... really 1/3, or something a little less?"