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Returning to recurrent state i.o.

A recurrent state is defined to be a state $i$ with $\mathbb{P}(\cup_{n=1}^{\infty}{X_n}|X_0=i)=1$. Now we wish to show $\mathbb{P}(\text{returning i.o.})=1$.

Define $A_m$ to be the event that the chain returns to $i$ at the $m^{\text{th}}$ time step and $B_m$ to be the event that the chain never returns to $i$ after the $m^{\text{th}}$ time step.

\[\mathbb{P}(\cup_{m=1}^{\infty} A_m \cap B_m)=\mathbb{P}(\text{returning finitely often})\]

So we wish to show

\[\mathbb{P}\left( \left(\cup_{m=0}^\infty A_m \cap B_m \right)^C \right) =1\]

which is the \mathbb{P}obability of returning $i.o.$. Now we see by axioms and the condition of a recurrent state.

\[\mathbb{P}(\cup_{m=1}^{\infty} A_m \cap B_m) \leq \mathbb{P}(\cup_{m=1}^{\infty} B_m)=0\]

Therefore we have the desired result.