Written: Sept. 12, 2015
The statement of the title, is, in fact, meaningless, because it tacitly assumes that we can add-up "infinitely" many numbers, and good old Zenon already told us that this is absurd.
The true statement is that the sequence, a(n), defined by the recurrence
a(n)=a(n-1)+9/10n a(0)=0 ,
has the finitistic property that there exists an algorithm that inputs a (symbolic!) positive rational number ε and outputs a (symbolic!) positive integer N=N(ε) such that
|a(n)-1|<ε for (symbolic!) n>N .
Note that nowhere did I use the quantifier "for every", that is completely meaningless if it is applied to an "infinite" set. There are no infinite sets! Everything can be reduced to manipulations with a (finite!) set of symbols.