I am trying to prove the following lemma 2.3.5 from Stochastic Processes, Sheldon Ross, 2nd ed, page 77 which I have provided it here for convenience. The proof is provided in the book based on a lemma before this lemma whose proof is very messy. I believe there must be a simpler proof for the main lemma which will not depend on the other lemma.
I am trying to prove it for $n=3$. I really appreciate any idea and help in this regard.