In the textbook Yeung, Raymond W: A First Course in Information Theory Chapter 2, the author makes the following proposition:
$X_1\rightarrow X_2\rightarrow...\rightarrow X_n$ forms a Markov chain if and only if
$X_1\rightarrow X_2\rightarrow X_3$
$(X_1, X_2)\rightarrow X_3\rightarrow X_4$
$(X_1, X_2, X_3)\rightarrow X_4\rightarrow X_5$
$...$
$(X_1, X_2,...,X_{n-2})\rightarrow X_{n-1}\rightarrow X_n$
and leaves the proof thereof as an exercise to the reader.
I have made various attempts at this proof (And deep-dives of maths.stackexchange) without success since I am lacking some fundamental understanding of how to formulate the proof.
Since the author leaves this as an exercise, I do not wish to solicit the answer, but I have no idea how to approach proving that this is indeed IFF and any conceptual guidance thereon would be appreciated.
As an added point, I am unsure if the following interpretation of the implied independence is correct:
$(X_1, X_2)\rightarrow X_3\rightarrow X_4$ is a Markov chain if
$(X_1, X_2) \perp X_4|X_3 $ which is true if
$p(x_1,x_2,x_3,x_4)p(x_3)=p(x_1,x_2,x_3)p(x_3,x_4)$
Any guidance/corrections here would very helpful.