Let $n$ be a natural number and $\{a_n\}$ be a bounded sequence of real numbers, that is $\vert a_n\vert\leq M$, for all $n$ ($M\geq0$). Define $E_n=\overline{\{a_j\vert j\geq n\}}$ as the closure of the set $\{a_j\vert j\geq n\}$. Then $E_n\subseteq[-M, M]$
As far as I know, the closure of a set can be defined as the set itself plus all its boundary points. Henceforth, since a boundary point can either belong or not to the set it bounds, I would say that $[-M, M]\subseteq E_n$ and NOT that $E_n\subseteq[-M, M]$.
A counterexample to $E_n\subseteq[-M, M]$ could be represented by a boundary point of the set $\{a_j\vert j\geq n\}$ which is not in the interval $[-M, M]$, that is which does not belong to the set it bounds.
Could you please clarify such a doubt and highlight the flaws of my reasoning?
Recall $A \subseteq B \implies \overline{A} \subseteq \overline{B}$ and $B = \overline{B}$ iff $B$ is closed.
Now, apply this with $A= \{a_j: j \geq n\}$ and $B = [-M,M]$.