Solution
= Solution
For a <Markov chain> started at state $i$, let $\tau_i^+=\inf\{n\geq1:X_n=i\}$ denote the first positive return time, using time $0$ for the starting state. A <recurrent state> has $P_i(\tau_i^+<\infty)=1$; a <transient state> has return <probability> strictly less than $1$. An <absorbing state> satisfies $P_{ii}=1$, so the chain remains there forever after arrival and it is necessarily a <recurrent state>.