The limit of a process at infinity is not necessarily subject to the same constraints as the outcome you're seeing after a finite number of steps. The assumption you're making here is intuitive, but it bites students in the butt every now and then.

Most simply, after "infinitely many" steps, the "arbitrarily close" in your mental model becomes "infinitely close", and in real numbers, "infinitely close" is the same as "equal to", because you don't have infinitesimals.

It's the same reason why we can construct irrational or transcendental numbers as a limit of an infinite series of rational numbers.

Well, consider my ass bitten ;) Your analogy to infinite series helps.