Sure, but you're equivocating two things that aren't the same. Until you've written infinity 9s, you haven't written the number yet. Once you do, the number you will have written will be exactly the number 1, because they are exactly the same. The difference between all the nines you could write in one thousand lifetimes and 0.999... is like the difference between a cup of sand and all of spacetime.

Or think of it another way. Forget infinity for a moment. Think of 0.999... as all the nines. All of them contained in the number 1. There's always one more, right? No, there isn't, because 1 contains all of them. There are no more nines not included in the number 1. That's why they are identical.