jammie wrote:

2.9¬ will always be 2.9¬ at no point is it 3.

Which of the maths did you do this in?

I assumed that we were talking about real numbers with the metric defined as d(x, y) = |x - y|, and that 2.9¬ was referring to the limit of the sequence

a_0 = 2

a_n = 9/10^n + a_(n-1)

(normally, I'd define it as a series using summation notation, but I don't think HTML can do that). The axiom of completeness applies to the reals, so the limit of a_n = a is a real number. Since for all e > 0, |a - 3| < e, doesn't this mean the limit is equal to 3 (unless I made a typo in the definition of the sequence)?