This little guy craves the light of knowledge and wants to know why 0.999... = 1. He wants rigour, but he does accept proofs starting with any sort of premise.
Enlighten him.
I'm being serious. The word 'infinity' refers to some points from spaces like Aleksandrov compactification. It does not refer to any sort of repetition of any pattern. You cannot, for example, write '0.999∞' or '0.999infinity' to mean the same thing as '0.999...' or '0.(9)'.