This has something to do with the geometric sum to infinity.
0.9999999... = 0.9 + 0.09 + 0.009 + ...
r(ratio of geometric progression) = 0.09/0.9 = 0.1
and a(the first term)
Sum of a geometric series approaching infinity is given as S=a/(1-r) if r<1, and S=a/(r-1) if r>1.
Therefore, given that r is less than 1...
S= 0.9/(1-0.1)
= 1
Convergence of numbers is when an total infinite number of sums is a finite number. 0.9 + 0.09 + 0.009 + ... will be 1 as the number of the sum approaches infinity.