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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 $A$

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  • $\begingroup$ 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. $\endgroup$ – epiliam Sep 10 '20 at 5:34
  • $\begingroup$ 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. $\endgroup$ – user810677 Sep 10 '20 at 5:40
  • $\begingroup$ I guess it comes down to the fact almost upper/lower bounds don't have any usage outside of $\lim \sup$ and $\lim \inf$. $\endgroup$ – user810677 Sep 10 '20 at 5:42
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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!

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

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