TITLE: Why do so few authors package up the definition of a limit into a function?
QUESTION [6 upvotes]: Occasionally, you will literally see people arguing for nonstandard analysis purely to unnest the quantifiers in the definition of a limit. By unnesting, I mean avoiding an exists quantifier that is required to be nested inside the scope of the for all epsilon quantifier, and instead having only foralls that can be made implicit and constraints on the corresponding variables. NSA does this by just constraining $N$ to be infinitely large or $x$ to be infinitely close to $x_0$.
But there's a much simpler way to clean things up so that the quantifiers are no longer nested. Just define functions:

A sequence $a_n$ in a metric space is convergent with limit L if there is a function $N : \mathbf{R^+} \to \mathbf{N}$ such that $d(a_n,L) < \varepsilon$ for all $\varepsilon > 0$ and $n > N(\varepsilon)$.
A sequence $a_n$ in a metric space is a Cauchy series if there is a function $N : \mathbf{R^+} \to \mathbf{N}$ such that $d(a_n,a_m) < \varepsilon$ for all $\varepsilon > 0$ and $n,m > N(\varepsilon)$.
A function $f : X \to Y$ between metric spaces has the limit $L = \lim_{x \to x_0} f(x)$ if there is a function $\delta : \mathbf{R^+} \to \mathbf{R^+}$ so that $d(f(x),L) < \varepsilon$ for any $\varepsilon > 0$ and $d(x,x_0) < \delta(\varepsilon)$.

which lets you write the definitions with only foralls that can be floated out to top level so that you just have one inequality between top level variables implying the other.
Furthermore, actually requiring that this function be defined (occasionally called the modulus of convergence), is both very convenient in epsilon delta proofs, and actually necessary to have for anything related to numerics/computability/constructive analysis or if you generally care about "how many terms do I need" questions.
So this made me curious. Is there any fundamental reason why you would want to write the definition of a limit in terms of nested quantifiers? I'd say tradition, but afaik first order logic is quite a bit newer than the definition of a limit. How did the 19th century analysts state the definition?

REPLY [2 votes]: I think your proposed answer of the 'problem' being a lack of focus on constructive mathematics is off the mark. At least in some scenarios, there are constructive differences between the alternating quantifier definitions and the ones using functions.
For your first two examples, consider the following propositions:

All constructive functions are continuous
Every continuous function from $\mathbb R$ to $\mathbb N$ is constant

I don't think restricting yourself to the positive reals changes 2. Anyhow, these two together indicate that your $N$ cannot actually depend on $ε$ constructively, which is unlikely to be what you want, I think.
To examine why the above works out the way it does, I think the third example is helpful. The traditional propositions-as-types way to interpret the ε-δ definition is like:
$$Π_{ε:\mathbb R}Σ_{δ:\mathbb R} ...$$
which is easily shown to yield:
$$Σ_{δ : ℝ → ℝ} ...$$
However, developments like homotopy type theory has shown that this interpretation is not as precise as it could be. This $Σ$ type is not a proposition, in the sense that all values are equivalent. So, to more precisely interpret the ε-δ statement, we should use the propositional truncation:
$$Π_{ε:ℝ}\Vert Σ_{δ:ℝ} ...\Vert$$
And it turns out that the principle:
$$ (Π_{ε:ℝ} \Vert Σ_{δ:ℝ} ...\Vert) → \Vert Σ_{δ : ℝ → ℝ} ...\Vert$$
is a genuine, non-constructive axiom of choice.
The thread tying the three examples together is $ℝ$ as a quotient type, and propositional equivalence allowing more freedom of choice than functions. In order for a function with type $ℝ → ℕ$ to respect the quotient structure of the real numbers, the choice of the natural number cannot depend on the detail of how the real number is approximated, and this ends up implying that the entire function is actually constant. However, all proofs of a proposition are considered equal regardless of the witness used to prove them; this is what the truncation is accomplishing, and in this case, I think it allows you to choose varying naturals based on the details of the approximation, in a way that is not valid for functions from $ℝ$ to $ℕ$. A similar thing can happen with the ε-δ statement, but since $ℝ$ is not discrete like $ℕ$, the discrepancy is more subtle.
I'm not really very knowledgeable on the exact methodology of constructive analysis. You could get around the above by stating things about approximants, rather than the quotient. But, if you want to be really precise about actually using $ℝ$, and stating the theorems in the usual way, you're going to have to be careful about using actual propositional statements, and paying attention to the difference between propositional $∀$-$∃$ and functions, even constructively. This shows up in other scenarios than HoTT, too, like realizability toposes, where interpretations of $∀$-$∃$ are less constrained than functions in general (although weak forms of the axiom of choice may hold, like countable choice).