Assume for a Markov Chain with period $d$, $\{C_0, C_1, \dots, C_{d−1}\}$ be the equivalence classes induced by $∼$ $(i$~$j$ means all the paths from $i$ to $j$ is of length $0$ mod $d$ )and numbered according to any $(i_0, i_0)$ cycle. If $(i, j)$ is an edge of G with $i \in C_k$, then $j \in C_{(k+1) mod ~ d}$. How to prove this ?
Actually my question is from a state $i$ why can't there be an edge to an equivalence class 2 or more step forward. I understand I have to use the property of equivalence class but not able to find the contradiction.