I'm now learning about almost upper/lower bounds and I find the choice of words very confusing.
Definition of $x$ = almost lower bound of $A$ :
$\{y \in A: y \geq x\}$ is finite.
What's strange about this definition is that it implies every upper bound of $A$ is also an almost upper bound of $A$. But that goes against the intuitive meaning of "almost". If x is "almost" y, then x is not y, but they are close. But in this definition x could be y.
So the definition should be $\{y \in A: y > x\}$ is finite and $\neq \emptyset$.
But then that doesn't work either because if $A$ is bounded and infinite, there might be no almost upper bounds, which intuitively makes sense but would mean $\lim \sup A$ doesn't necessarily exist.
I think my confusion comes from the word "almost". I think a better word would be "partial" or "sub". Because if an upper bound of A is $\geq$ every element in $A$, then of course it is $\geq$ every element in a part of $A$, or a subset of $A$
Weird side point, lower bounds don't imply anything about upper bounds.
If $A$ is infinite and the set of all members which can be described as less than or equal to $x$ is a finite set - then its compliment, (the set of all members which can be described as greater than $x$), is an infinite set like $A$. That compliment is "almost $A$" and $x$ would be its lower bound.
So, $x$ isn't the lower bound of $A$, but $x$ is useful in defining subsets of $A$ and understanding its composition.
Hope this helps!