Show that if a Markov chain is irreducible and has a state $s_i$ such that $P_{ii}>0$, then it is also aperiodic.
Since $X$ is irreducible, for any state $j$ there exist positive integers $n,n'$ such that $P_{ij}^n>0$ and $P_{ji}^{n'}>0$. Since $P_{ii}>0$, it follows that $P_{ii}^m>0$ for all positive integers $m$, and hence $$P_{jj}^{n+n'+m}\geqslant P_{ji}^{n'}P_{ii}^mP_{ij}^n>0. $$ This implies that the period of state $j$ is $1$. Since $j$ was arbitrary, we conclude that $X$ is aperiodic.