Suppose that $I$ is an index set and $f$ is a function that maps indices into numbers (or, more generally, to elements of an ordered set).
The definition of supremum has two clauses.
(1) $$f(i)\le \sup_{j\in I}f(j)$$ for all $i\in I$.
(2) If $f(i)\le s'$ for all $i\in I$, then $$\sup_{i\in I}f(i)\le s'$$.
Clause (1) bounds the supremum from below, and clause (2) bounds it from above.
Lemma 1. If $A\subseteq B\subseteq I$, then $\sup\limits_{i\in A}f(i)\le\sup\limits_{i\in B}f(i)$.
Proof. For all $i\in B$, including $i\in A$, we have $f(i)\le\sup\limits_{i\in B}f(i)$ by (1); therefore, $\sup\limits_{i\in A}f(i)\le\sup\limits_{i\in B}f(i)$ by (2).
Now I rewrite Opalg's proof in more detail.
Lemma 2. $\sup_{n\in\Bbb{N}}\And[O(x,n)] \le \And[O(x,\infty)]$.
Proof. $O(x,n)\subset O(x,\infty)$; therefore,
\[
\And[O(x,n)]=
\sup\limits_{a,b\in O(x,n)}d(a,b)\le
\sup\limits_{a,b\in O(x,\infty)}d(a,b)=
\And[O(x,\infty)]\qquad(3)
\]
by Lemma 1. In applying the lemma, we instantiate $I$ with $M^2$ (i.e., the set of all ordered pairs $\langle a,b\rangle$ where $a,b\in M$), $A$ with $O(x,n)^2$, $B$ with $O(x,\infty)^2$, and for $i=\langle a,b\rangle\in M$ we set $f(i)=d(a,b)$. Now by applying clause (2) of the definition above to (3), we get
\[
\sup\limits_{n\in\mathbb{N}}\And[O(x,n)]\le \And[O(x,\infty)],
\]
as required. In applying (2) we set $I=\mathbb{N}$ and $f(i)=\And[O(x,i)]$ for $i\in\mathbb{N}$.
Lemma 3. $\And[O(x,\infty)]\le\sup_{n\in\Bbb{N}}\And[O(x,n)]$.
Proof. As Opalg explains, for all $a,b\in O(x,\infty)$ there exists an $n\in\Bbb N$ such that $a,b\in O(x,n)$. Therefore, for all $a,b\in O(x,\infty)$ we have
\begin{align}
d(a,b)&\le\sup\limits_{a,b\in O(x,n)}d(a,b)&&\text{by (1)}\\
&=\And[O(x,n)]\\
&\le\sup_{n\in\Bbb N}\And[O(x,n)]&&\text{by (1)}
\end{align}
Now we apply (2) to $d(a,b)\le\sup\limits_{n\in\Bbb N}\And[O(x,n)]$ and get
\[
\sup_{a,b\in O(x,\infty)}\le\sup_{n\in\Bbb N}\And[O(x,n)].
\]
The required statement is the conjunction of Lemmas 2 and 3.