Since and is continuous with , letting gives .

For directly: .

Alternatively, conditioning on and writing for independent copies of the process started from one individual,

By induction: , and assuming ,

so and taking expectation gives .

Let be the smallest non-negative solution of . Induction on shows : the case is clear, and increasing on gives

Hence as , while because itself solves the fixed point equation; so .

For the criterion that exactly when , first suppose . Then and , so

and since , necessarily .

Now assume and . Define , so and

because some with . Hence is strictly increasing on , so by Rolle’s theorem has at most one root other than .

If there is no other root then , and since and we have on , whence

that is .

If instead for some , that root must be the extinction probability . Rolle’s theorem gives with , and strict increase yields

so .