= Solution
A finite <irreducible Markov chain>[irreducible] <aperiodic Markov chain> converges to its unique <stationary distribution> from every initial state. Finiteness holds because nonnegative entries with fixed finite row sums are <bounded>. The given connecting-sequence property makes the chain <irreducible Markov chain>[irreducible]: each feasible step in such a sequence has positive proposal probability and is accepted. The all-zero index table occurs with probability $1/L>0$ and gives a self-loop at every state, establishing <aperiodicity>. Reversibility supplied the stationary uniform law in (b). Consequently
$$
\boxed{\Pr(X_t=x)\longrightarrow\pi(x)\quad\text{for every state }x.}
$$
The conclusion also holds when the feasible state space is a singleton.
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