In Proposition 5.4.5 of Meyn and Tweedie's Markov Chains and Stochastic Stability, it is said that if a chain $\Phi$ is $\psi$-irreducible and aperiodic, then every $m$-skeleton of it is also $\psi$-irreducible and aperiodic. The authors did not give an explicit proof of this.
It is easy to show aperiodicity using induction, but I had a hard time to figure out the proof for $\psi$-irreducibility. Is induction and contradiction a right direction to go? In general, if there is no aperiodicity of the chain, is it also true that all the $m$-skeletons are $\psi$-irreducible?