 Solution

For a Markov chain started at state $i$, let ${\tau }_i^{+}=\inf \{n\ge 1:X_n=i\}$ 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.