I have to solve the following question:
Consider a random walk Markov chain on $S = \{1, 2, \ldots, 100\}$. If the chain is between 2 and 99, it selects one of the adjacent states with equal probabilities. If the chain is in state 1 or 100, it either stays in this state with probability 0.5 or moves to the adjacent state (with probability 0.5). Prove the following as though you have never heard of irreducibility and aperiodicity.
$\mu^{(n)} \xrightarrow{TV} \pi$ for any choice of $\mu^{(0)}$.
I don't see how to do this, otherwise than to work out every step in the proof of the equilibrium theorem where irreducibility or aperiodicity is stated.
Could you please give me a hint on where to start?