When we are introduced to $\epsilon-\delta$ proofs in a usual first-year introductory course to Calculus, we usually always tend to do one of two things when we attempt to prove limits using the $\epsilon-\delta$ definition of a limit.
When we attempt to prove a limit of the form :
$$\lim_{x \ \to \ a} f(x) = L$$
- We show a relation between $\epsilon$ and $\delta$ by choosing a value of $\delta > 0$ equal to $\epsilon$, (how we choose this value depends on how we work our way towards that point in out proof). Afterwards we then have to show that the chosen value of $\delta$ satisfies the inequality $0<|x-a|<\delta \implies |f(x)-L|<\epsilon$ , $\forall \epsilon > 0$
- We write delta as a function of epsilon $\delta(\epsilon)$ since, delta is dependent on epsilon, and by that we can (arguably more intuitively) show that $\forall \epsilon >0 \ (\exists \ \delta>0 \ni (0<|x-a|<\delta \implies |f(x)-L| < \epsilon))$
But in either of these cases, we prove the existence of a single $\delta$ for a given $\epsilon$, however multiple $\delta$'s can exist for a single given $\epsilon$, but we tend to disregard that fact in our proofs as the existence of a single \delta satisfies the formal definition of a limit. Therefore the set of $\delta$'s that we prove exists (by proving every $\epsilon > 0$ has a single corresponding $\delta >0\ $) is only a small subset of the set of all possible $\delta$'s satisfying the inequality ($0<|x-a|<\delta \implies |f(x)-L|<\epsilon$ , $\forall \epsilon > 0$)
If we let $S$ be the set of all possible $\delta$'s satisfying the inequality :
$$S = \{\delta \in \mathbb{R} \ : \ 0<|x-a|<\delta \implies |f(x)-L|<\epsilon\}$$
All we do in our $\epsilon-\delta$ proofs (in first-year) is show that the value of $\delta$ we choose or write as a function of epsilon $\delta(\epsilon)$ is an element of that set $S$, and by showing that the inequality holds with the $\delta$ we've chosen for a given $\epsilon$. we are essentially showing that our chosen $\delta$ lies between the Infinum of $S$ and the Supremum of $S$.
So in a nutshell in our first-year $\epsilon - \delta$ proofs all we show in our proofs is this about our chosen value of $\delta$ :
- $\delta \in S$
- $inf(S) \leq \delta \leq sup(S)$
Which are both logically the same thing. But what about the other values of $\delta$ for a given $\epsilon > 0$? How can we find the set $S$ of all possible $\delta$'s for a given $\epsilon > 0$? How do we find, $inf(S)$ and $sup(S)$?
Lastly am I correct in everything that I have said above about $\epsilon-\delta$ proofs?