Solution
= Solution
The convergence theorem for an irreducible aperiodic positive recurrent <Markov chain> says $\mathbb P_i(X_n=j)\to\pi_j$ for every pair of states. Apply it to $i=j=0$ and take the complement:
$$
\boxed{\lim_{n\to\infty}\mathbb P_0(X_n\ne0)=1-\pi_0=\frac1{1+p}.}
$$
Although $p_{00}=0$, the chain is aperiodic because its possible return lengths have greatest common divisor one; a self-loop is sufficient but not necessary for <aperiodicity>.