0.999999999... indeed can be 1. Same as 0.33333... is equal to 1/3. Though you can see it's a fraction. Same thing is basically with 0.99999... to 1. It's just the fact that the 1 is a whole number but the 0.9999... looks messy. The terms of the associated sequence 0.9, 0.99, 0.999,..., do "get arbitrarily close" to 1, in the sense that, for each term in the progression, the difference between the term and 1 gets smaller and smaller. The number 0.999... (with the ellipse) means that it goes on forever, there is no stop to it meaning that it basically is, and has been proven, that it is 1.
