That's a good argument. I think I'm beginning to see where you're coming from - it's something that most of the people on the nines-recurring-equals-one side brush over, but one really shouldn't be so blasé about throwing infinities around. Recurring fractions don't seem to cause a lot of problems, but an amateur mathematician should be aware of the potential for mayhem in the original problem statement, whether or not they are reassured by any of the arguments.