Is O(n) a proper class or a set? Is $O(n)$ as the collection of all functions that are bounded above by $n$ a proper class or just a set?
What about $O(\infty)$?

There are multiple possible ways to define it, but on reasonable way to define it is like so:
$O(g(n))$ is the set of all functions $f:\mathbb{R}\to\mathbb R$ where there exist $k, N\in\mathbb R$ such that whenever $n>N$, $f(n)<k g(n)$. (i.e. $f$ is eventually bounded above by $g$)
That means $O(n)$ is the set of functions that are bounded above by the function $g(n)=n$.
$O(\infty)$ would be the set of all functions $\mathbb R \to \mathbb R$, since everything is bounded above by $\infty$.
