# Notation: for $x$ not much less than $a$

"Everybody knows" that $a\ll b$ means a quite vague thing, something like $a$ is very much less than $b$.

(And on math.stackexchange.com, it may be observed that not everybody knows the difference in MathJax and LaTeX code between $a\ll b$ and $a<<b$.)

My question is whether there is a conventional notation for a similar concept, but which I will define precisely below, and if there's not, then what would be a good notation for it? For now I'll use the notation $x\preceq a$.

Its precise definition is this: $$P\text{ holds for }x\preceq a \tag 1$$ $$\text{means}$$ $$\text{for some }\varepsilon>0,\text{ for all }x\in(a-\varepsilon,a),\text{ P holds.}$$ One could express this as saying $P$ holds for $x$ not much less than $a$, and this gives a precise definition to that concept. But notice that $x$ and $a$ do not play symmetrical roles but with the direction of the inequality reversed, i.e. this does not mean the same thing as "$P$ holds for $a$ not much more than $x$". Probably it would be a good idea to have a reminder of that asymmetry in the notation. And of course I'd like to keep it simple.

So we want something

• as short and simple as line $(1)$ above (in particular, temporarily sweeping under the carpet all attention to the quantity $\varepsilon$ and the quantifiers $\forall$ and $\exists$), but
• with the needed suggestive asymmetry, and
• not too easlily confusable with other frequently seen conventional notations that have different meanings.
• Now I'm leaning toward writing $x\prec\prec a$ . . . . Dec 8, 2013 at 20:27
• Now I think maybe the best option is to write "$P\text{ holds for }x\uparrow a$" or "$P\text{ holds for }x\downarrow a$" or "$P\text{ holds for }x\to a$", as the case may be. Dec 17, 2013 at 19:23

We could write $$P \text{ holds } \forall x \in {}_{\varepsilon}{a}$$ to mean

for some $\varepsilon>0$, for all $x\in(a-\varepsilon,a)$, $P$ holds.

Of course it makes sense to define the same notion on the right: write $$P \text{ holds } \forall x \in a_\varepsilon$$ to mean

for some $\varepsilon>0$, for all $x\in(a,a+\varepsilon)$, $P$ holds.

• I'm not sure this is worth doing unless it's really a lot simpler than writing out "for some $\varepsilon>0$, for all $x\in(a-\varepsilon,a),\text{$P$holds}$." Your notation still seems too complicated. Dec 8, 2013 at 20:25
• Maybe I didn't get your point, but to write "$P$ holds $\forall x\in {}_\varepsilon a$" is much shorter than "for some $\varepsilon>0$, for all $x\in(a−\varepsilon,a)$, $P$ holds." Dec 8, 2013 at 20:51
• Shorter in actual length, yes. Maybe what I want is a notation that temporarily sweeps under the carpet all attention to $\varepsilon$. Dec 8, 2013 at 21:00
• Ok, then I didn't get it. Maybe you should write this requirement explicitly in your question ;-) Dec 8, 2013 at 21:01
• Now you're bringing to my attention another desideratum: I'd like the reader to be temporarily distracted from the whole issue of the precise meaning so that while reading "$P$ holds for $x\preceq a$", the attention would be more on the meaning of $P$ (which might in itself be fairly involved) and on $x$ being a little bit less than $a$ than on just what that last thing means. If the reader sees ${}_\bullet a$, the reader will be thinking "I remember: he said that means thus-and-so". A sort of mental speed bump---just what I want the notation to avoid. Dec 8, 2013 at 21:08

How about "$P$ holds in some open left-neighborhood of $a$"?

I think this even agrees with the definitions used in some calculus courses.

• Well, again, that puts a lot of words into the point I'd like to temporarily distract the reader from. I have in mind an audience of non-mathematicians who are perfectly fluent with mathematical calculations and elementary algebra, such as happen in (for example) finding $\int\sec^3 x\,dx$, but less so with logical reasoning such as happens in proofs. Dec 8, 2013 at 21:12
• $P$ holds for all $x \in N^-(a)$? (Obviously, $N^-(a)$ is not really a set since it depends on a hidden $\epsilon$ which is implied to exist, so it might be weird notation in some ways, but maybe convenient in others.) Dec 8, 2013 at 21:17
• Another possible option is "$P$ holds for $x \to a^-$", which is a notation you can write informal calculus in. For example "$x^2 \approx 4$ for $x \to 2^-$". Dec 8, 2013 at 21:21

Is it more the less-than aspect that is key, or the nearness? I wonder if

$$x \mathbin{{\approx}_{-}} a$$

might work, by analogy to the $\lim_{x\to a^{-}}$ operator, though this highlights the nearness more than the ordering.