This is correct. For example: Let's say the population is P and the net reproduction rate (the number of daughters a woman has in her lifetime) is 0.9. The next generation would have 0.9P people, the next one would have 0.9², then 0.9³, etc. Even with immortality you would get lim_(i→∞)∑ᵢ0.9ⁱP = 9P. So as long as the net reproduction rate is less than one the population stabilises at a finite number even if everyone is immortal.

> As long as the fertility rate is less than 2 children/woman, the total population will still converge even if no one dies.

Converge to what? How?

I think they mean the infinite sequence of

Σ (generation_n)

always converges to some finite number when generation_(n+1) = k * generation_n has k < 1, which is equivalent to <2 immortal children per woman.

I don’t think that’s a good model, as people who want more offspring will out-reproduce those who don’t, but that’s an argument for another time.

> I don’t think that’s a good model, as people who want more offspring will out-reproduce those who don’t, but that’s an argument for another time.

This argument assumes the offspring behaves in the same way as their parents. I don't see that necessarily being true. Even if that is the case, there are legislative solutions.

To population 0. 1.9 children / woman is a .95 multiplier on human population per generation, like a -5% interest rate. Assuming birth rate is constant < 2, it converges to 0.

Not "if no one dies" which was part of GGP's claim.

Correct, my mistake. Other comments with summations are the accurate convergence then.