How to interpret an event of a Markov chain? Suppose $\{X_n\}_{n=0}^{\infty}$  is a Markov chain with state space $S = \{0,1,2,...,N\}$ with
$$
P(X_1=0|X_0=0)=1
\\
P(X_1=N|X_0=N)=1
$$ then why the following result is true?
$$
\{X_n=N\}\subset{\{X_{n+1}=N\}}
$$
In fact, I am having hard time to understand how does the event $\{X_n=N\}$ look like. What are the elements of the event $\{X_n=N\}$?

What this result says is that if $X_n = N$ then $X_{n+1} = N$, which is true since $P(X_{n+1} = N | X_n = N) = 1$, as you state.
Regarding your second question, the event $X_n = N$ consists of all infinite sequences $X_0,X_1,X_2,\ldots$ satisfying $X_n = N$.
