Currently I am reading this Markov Chains' notes. At page $112,$ section $4.2$ Transition matrix, I fail to understand how author obtains the following.
As seen above, the random evolution of a Markov chain $(Z_n)_{n\in \mathbb{N}}$ is determined by the data of $$P_{i,j} := P(Z_1 = j | Z_0 = i), i, j \in \mathbb{S},$$ which coincides with the probability $P(Z_{n+1} = j | Z_n = i)$ which is independent of $n \in \mathbb{N}.$
I do not understand how author obtains the following equality $$P(Z_1 = j| Z_0 = i) = P(Z_{n+1} = j| Z_n = i).$$ I think Markov property might be applied here, but I do not know how.
Any hint is appreciated.
You cannot prove this. Many books consider what are called homogeneous Markov chain. For these what you are trying to prove is part of the definition. There are non-homogeneous MC's so Markov property does not imply this property.