Why is an almost upper bound named so?

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 $$A$$

• There is not always a direct relation between the English meaning and the mathematical meaning. For example, sets can be both closed and open, or, neither. In this example I think its fine as the set of almost bounds is a superset of the set of bounds. – epiliam Sep 10 '20 at 5:34
• You have a point but $\mathbb{R}$ is also a superset of the set of bounds. It'd be ridiculous to define an almost upper bound to be another term for a real number. – user810677 Sep 10 '20 at 5:40
• I guess it comes down to the fact almost upper/lower bounds don't have any usage outside of $\lim \sup$ and $\lim \inf$. – user810677 Sep 10 '20 at 5:42

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.
• Ah so technically $x$ is the lower bound of almost $A \implies x$ is the almost lower bound of $A$. I see the connection. – user810677 Sep 10 '20 at 5:47