# The (epsilon-delta) tag.

The currently does not have a tag wiki.

Can someone propose one?

Below is why I ask the question

Motivation 1: every proof in hard analysis is, in some sense, an $\epsilon-\delta$ proof. So I don't really see the point of such a tag. If someone can point out a more restrictive usage of this term that would be nice.

Motivation 2: I've seen a lot of beginning students use the phrase "$\epsilon-\delta$" to refer to any rigorous proof, possibly because their first introduction to mathematical proofs arise in the context of a course that uses, for example, Baby Rudin as a textbook. If that is indeed the case this tag smells rather like a meta-tag to me.

• I propose that we make it so that we don't have to worry about a tag wiki. – Asaf Karagila May 14 '14 at 14:57
• Well, there might be question, where the student is asked to compute a limit/show continuity directly using $\varepsilon$-$\delta$ definition. As opposed to questions, where he is allowed to use results that were already proven. (I.e., it is allowed to use other things too, not just definition.) – Martin Sleziak May 14 '14 at 15:44
• @MartinSleziak: so you are thinking in terms of it being a refining tag of calculus and algebra-precalculus? – Willie Wong May 14 '14 at 15:55
• To be honest, the only 20ish people that are actually using this tag didn't make the impression that they use $\varepsilon-\delta$ for any kind of rigorous analytical proof, as Motivation 2 implies. At least not on MSE, that is. – chiru May 15 '14 at 2:30
• @chiru: I just realised after reading your comment that my question is unclear. The two motivations are the motivations that drove me to ask this question, not my suppositions of how the tags are used. In fact I ask this question precisely because I have absolutely no idea why such a tag would be at all useful. – Willie Wong May 15 '14 at 7:32
• I remember when I did my first course on analysis, and the $\epsilon-\delta$ proofs bamboozled me (and the entire class!). So the then-me would have used this tag as Martin Sleziak prescribed. I think that that is what the tag should be for (that is, people struggling with this method of proof, not with a specific proof). If noone is using it for that, then I would think it is superfluous. – user1729 May 15 '14 at 8:38
• (For example, the three most recent post on this tag are precisely what I mean [1], [2], [3]. The questions are about issues with the method of proof, the actual proof is a side-issue.) – user1729 May 15 '14 at 8:44
• @user1729: I don't exactly understand. My impression of those three questions are "how do I give an explicit relation $\delta = \delta(\epsilon)$." Is that what you mean? – Willie Wong May 15 '14 at 8:53
• @WillieWong I suspect my answer is "yes", but your bit in quote marks doesn't make sense...I mean that the tag would not be for a hard proof, but for a basic, low-level introductory one. Like if I was to ask "Prove that every finite $p$-group has non-trivial centre using induction" then I wouldn't use the induction tag - the issue is with the group theory not the method of proof. – user1729 May 15 '14 at 8:56
• @user1729: sorry, edited. I guess you would support a case where the use of the induction is less trivial, for example instead of $p-1 \implies p$ you have something between usual induction and strong induction, like $\lceil p/2 \rceil \implies p$. – Willie Wong May 15 '14 at 9:07
• @WillieWong Yeah - questions where the actual induction is the issue, not the ambient subject (such as group theory or number theory or whatever). Am I making sense? – user1729 May 15 '14 at 10:41
• @user1729: I'll write up a proposed tag wiki and post it below as a CW answer based on your comments. Please check back in about 10 - 15 minutes and see if you agree. – Willie Wong May 15 '14 at 10:42

A proposed tag wiki excerpt based on user1729's comments above:

Questions about the details of implementing an $\epsilon$-$\delta$ proof, which typically appears in the context of analysis; for example about how the appropriate $\delta$ can be chosen for a given $\epsilon$.

A proposed tag wiki:

In mathematical analysis one often encounter statements of the form "For every $\epsilon > 0$, we can find $\delta > 0$ such that ... holds." Some examples include:

• The statement of continuity of functions defined on metric spaces: A function $f:X\to Y$ where $(X,d_X)$ and $(Y,d_Y)$ are metric spaces is said to be continuous at a point $x\in X$ if for every $\epsilon > 0$ we can find $\delta > 0$ such that for every $z\in X$ satisfying $d_X(x,z) < \epsilon$, we have that $d_Y(f(x),f(z)) < \delta$.
• The statement of convergence of series of real numbers: A series $\sum_{n = 1}^{\infty} x_n$ of real numbers $x_n\in \mathbb{R}$ is said to converge to $s\in\mathbb{R}$ if for every $\epsilon > 0$ there exists $N_0\in \mathbb{N}$ such that for every $N \geq N_0$ we have the partial sums $$\left| s - \sum_{n = 1}^N x_n \right| < \epsilon$$

Use this tag to ask questions about verifying that such a statement holds for a particular mathematical object; in particular about the process in which one comes up with a suitable "$\delta$" based on the given $\epsilon$.

• Perhaps the "specifically" could be replaced with "for example", or "mostly", or something else less definite? – user1729 May 15 '14 at 11:05
• @user1729: I made the post CW for a reason. ;-p – Willie Wong May 15 '14 at 11:13
• Okay, I made the "in particular" into "for example". I am less sure about my second edit: I also changed the "specifically" into "in particular". I didn't want to use "for example" twice, and felt that "in particular" is not absolute (unlike "specifically"). – user1729 May 15 '14 at 11:31