determining if a tail event I am to determine if $$\{\sup X_n < \infty \}$$ is a tail event, the solutions are as follows:

I don't understand how they got the line of equalities, specifically the last one, and why it holds for all $M$ and $n$.
 A: First, why is it true that
$$
\tag{1}
\left\{\sup_{n\in\mathbb{N}} X_n < \infty\right\}
= \left\{\sup_{n\in\mathbb{N}} X_{n+m} < \infty\right\}
$$ for all $m \geq 0$?
Fix $m \geq 0$.
If $\sup_{n\in\mathbb{N}} X_n < \infty$, then there exists some $C$ (a real-valued random variable) such that $X_n \leq C$ for all $n \in \mathbb{N}$. In particular, $X_{n + m} \leq C$ for all $n \in \mathbb{N}$, so $\sup_{n\in\mathbb{N}}X_{n+m}<\infty$. This shows that
$$
\left\{\sup_{n\in\mathbb{N}} X_n\right\}
\subseteq \left\{\sup_{n\in\mathbb{N}} X_{n+m} < \infty\right\}.
$$
Conversely, suppose $\sup_{n\in\mathbb{N}} X_{n+m} < \infty$.
Then there exists a real-valued random variable $C$ such that $X_{n+m} \leq C$ for all $n \in \mathbb{N}$. We can now define
\begin{equation*}
D = \max\{X_0, X_1, \ldots, X_m, C\}
\end{equation*}
so that $X_n \leq D$ for all $n \in \mathbb{N}$, and hence $\sup_{n\in\mathbb{N}}X_n < \infty$. This shows that
$$
\left\{\sup_{n\in\mathbb{N}} X_{n+m} < \infty\right\}
\subseteq \left\{\sup_{n\in\mathbb{N}} X_n\right\}.
$$
Thus, $(1)$ holds.

Next, why is it true that
$$
\tag{2}
\left\{\sup_{n\in\mathbb{N}} X_{n+m} < \infty\right\}
= \bigcup_{M=1}^\infty \bigcap_{n=1}^\infty \{X_{n+m} < M\}
$$
for all $m \geq 0$?
Note that for a sequence of real numbers $(x_n)_{n\in\mathbb{N}}$, saying that $\sup_{n\in\mathbb{N}} x_n < \infty$ is equivalent to saying that there exists an $M \in \mathbb{N}$ such that $x_n \leq M$ for all $n \in \mathbb{N}$ (i.e., the supremum being finite is equivalent to the sequence being bounded above by a positive integer). Fix $m \geq 0$. For each $\omega$ in the sample space of the random variables, $(X_{n+m}(\omega))_{n\in\mathbb{N}}$ is a sequence of real numbers, so, as above, saying that $\sup_{n\in\mathbb{N}} X_{n+m}(\omega) < \infty$ is equivalent to saying that for some $M \in \mathbb{N}$ we have $X_{n+m}(\omega) \leq M$ for all $n \in \mathbb{N}$. Translating this into the language of events, we have $\omega \in \left\{\sup_{n\in\mathbb{N}} X_{n+m} < \infty\right\}$ if and only if $\omega\in\bigcup_{M=1}^\infty \bigcap_{n=1}^\infty \{X_n \leq M\}$. This proves (2).
