Markov chains: An issue in classification of states

Question

I recently came across a lemma which goes as follows.

Suppose a Markov chain has N states. Let i and j be pair of states. Then j can be reached from i iff there is an integer $ 0 ≤n< N$ such that $p_{ij}^{n}>0$.

Is the inequality necessary at all here ($n<N$)? If the chain is finite and if there is a integer such that $p_{ij}^{n}>0$ can't we say that j can be reached by i without considering the number of states?

Answer 1

"Is the inequality necessary at all here ($n<N$)?"

Yes, since you have N states, then the longest way (if it exists) will be $N-1$.

If the chain is finite and if there is a integer such that $p^n_{ij}>0$ can't we say that $j$ can be reached by $i$ without considering the number of states?

Yes. The main problems here are absorbing states which will not allow you to go somewhere else and disconnected graphs but in those cases $\forall n\in [0, N-1], p^n_{i,j}=0$.