Here's another way to think about why .9999999999999 (repeated indefinitely) = 1
Suppose it doesn't: then 1 - .9999r must be > 0. But I can demonstrate that for any difference you pick, I can get it closer to zero than that. Ergo, the difference must be zero.
I have an undergrad degree in math. A lot of it comes down to the fact that things involving infinity (in this case, an infinite sequence) don't behave in ways that are particularly intuitive. That's what homework was for.
