That you can take the infinite and define it? Infinite to finite. That seems like human presumption to something for which we have no real answer. But then, we made numbers, so we can define them.
When did we determine the value of infinity?
What is the "limit of the sequence?"
Forgive me if my unwillingness to accept this grows so obtuse that it annoys people, I simply want someone to show me why I should accept something that appears, by definition as infinite, as being absolute.
I'm not a huge WIKI fan, but if you read further, you will also learn that there are no real numbers, they're all irrational, which arguably makes this whole issue inane.
So basically, 1 is 1. 9 is 9. 9/9 is 1. .999 repeating is 1, according to responses here. But what is 1? What is 9? Human construct? How does .999 repeating reach 1, since it's defined as continuing on into infinity, other than as the human construct designed to provide an absolute when we really have none? Inability to grasp the totality (or absence of totality) of our existence?
Are we arguing that .999 repeating is 1 for the sake of human sanity, the inability to see beyond infinity, the result of it being the creation of a construct that we define in terms of our own finite existence, or convenience because we cannot actually define it and at some point it was deemed prudent to make the assumption and close out infinity to support whatever longevity humanity has in the universe? Is infinity, then, the figment of human imagination?
Infinitesimal might be the best guiding word I can find to support your argument and somehow concede that our definition of specific numeric designations is....relevant. Otherwise, based on human definition of infinity, I cannot see how something that continues on into the infinite can be defined as finite (especially if you consider that it never reaches the point of the finite by definition of human construct).
I've been drinking, so I'm gonna shut up. Hopefully this somehow reflects my argument. If not, I'll try to reprise it tomorrow so that it offers some relevant rebuttal. If the responses continue in such an overwhelming fashion, I'll probably just shut up and admit I was wrong about the proof.