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"Immigration as a solution for fertility" is rooted in the assumption that the developing world stays forever poor and keeps producing surplus fresh meat for you to attract and assimilate, which is some combination of offensive, racist, and also deeply incorrect.

or be more creative. think of technological advancement so rapid that a less numerous generation can still be more productive than the one that preceded it.

That is valid, but I don't know if you can still talk of "society" about a very reduced population, it depends on the reduction factor and stops at reproduction rate zero.

A probable outcome would be that another society takes over the one disappearing.

Immigration or population replacement?

Immigration is the base of any country.

Your may be talking about more strict immigration policies but the subject here is "societal collapse" and population reduction and I was pointing out a possibility.