For a birth and death process, prove that $\lim_{n\to \infty}P(X_n=0) $ exists.

Question

I found this question in a book, and I do not know how to prove it. The author claims the following:

Consider a populations capable of generating children. Let $X_n$ be the number of individuals in generation $n$. Prove that $\lim_{n \to \infty} P(X_n =0)$ exists.

How would one prove such existence?

Answer 1

Let $n \in \mathbb{N}^*$, then,

$$(X_{n} =0) \subset (X_{n+1} = 0)$$

Indeed if you have no individuals at generation $n$ then you have no individuals at generation $n+1$.

Therefore, using the properties of increasing sequence of events we have that $\lim_{n \to \infty} P(X_n=0)$ exists and moreover,

$$\lim_{n \to \infty} P(X_n=0)=P\left(\bigcup_{n=1}^{+\infty} \ (X_n=0) \right)$$