Magnus Anderson wrote:There are infinite sums that evaluate to finite numbers. The argument put forward by JSS and people on his side is that \(0.9 + 0.09 + 0.009 + \cdots\) is not one of them.

As I say, due to JSS's extreme prolificness (if I may coin a word) on this site, plus the fact that he's not here to defend his ideas, I prefer not to argue with him due to basic fairness.

So if you could frame your points from the beginning and not ascribe them to JSS, then I won't be in a position to try to understand the ideas of JSS, which I found faulty four years ago. Just explain your ideas to me in your own terms. Else I'm arguing with someone who can't argue back.

Again, as I've said many times, if you would consult a book calculus or real analysis, you would know that .999... represents the geometric series 9/10 + 9/100 + ... whose sum, as defined in real analysis, is 1. I know you have an infinite series on one hand and a number on the other, but they are indeed equal mathematical objects, and this can be rigorously proved from first principles.

Magnus Anderson wrote:No no no, it's not phyllo who's uncomfortable. Phyllo is on your side. It's us. Those who deny the popular belief. We are the ones who are uncomfortable and only allegedly so.

I do desire to understand the nature of your disagreement. But as you keep falling back on claiming that the standard definition of a limit is a lie, without providing any more supporting details, I remain puzzled. The definition of a limit is what it is, as is the way the knight moves. Rules in a formal game. They can't be right or wrong, they're just formal rules that have turned out to be interesting and (in the case of math) useful in understanding the world.