Let $(X_n)$ a Markov chain on $N_0$ with $p_{00}=1$ and $P_i (\cup_{n=1}^\infty \{X_n=0\})>0$ Show $P_i (\cup_{j\in N} \{X_n=j \, i.o.\}) =0 \forall i\in N_0$
Intuitively this is true, but I can not proof it. My first thought was Borel-Cantelli but that does not work. How is this done correctly ?
Because $p_{00}=1$, $\{0\}$ is a closed class. Now $P_i(\bigcup_n\{X_n=0\})>0$ means the communicating class containing $i$ is open for all $i\neq 0$, so cannot be recurrent (as recurrent classes are necessarily closed). Hence the result follows (assuming your index set $N$ in $P_i(\bigcup_{j\in N}\{X_i=j\text{ i.o.}\})$ does not contain $0$).