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What conceptual difficulties do students learning epsilon-delta proofs have, or, why are the proofs difficult?

Motivation

I have to teach people and never found the epsilonistics particularly difficult, so I have trouble relating to the students and understanding their difficulties.

Desirable answers

  • Personal experiences
  • References to essays or academic literature
  • What students have told you

I am not interested in

  • Advice to use a particular method of teaching
  • Advice to use a particular book in teaching
  • Arguments about the desirability of teaching epsilonistics
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Order of quantifiers is always confusing; having to do bookkeeping is not a pleasant task either. –  Asaf Karagila Aug 24 '12 at 12:29
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I don't think it's the $\varepsilon, \delta$ which is difficult for many beginners in math. It was my impression that's more the logic operators $\forall, \exists$, their relation to each other in a formal statement and negation of these. Also this kind of reasoning is close to if not the first encounter with mathematical rigor, which you have to get used to. (of course this then applies to the $\varepsilon, \delta$ stuff in particular ;-) –  user20266 Aug 24 '12 at 12:38
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The intellectual difficulty of logical reasoning increases with quantifier depth. In verbal/mental shorthand, we tend to reduce the depth, e.g., by saying "x goes to 0," which is not the same as $x\rightarrow0$. Often these informal shorthands can be interpreted as valid statements about infinitesimals, either in their informal 18th-century incarnation or in NSA. (NSA enthusiasts claim that NSA usually eliminates one level of quantification compared to epsilon-delta.) We're mentally and verbally switching back and forth through all these levels, and it gets confusing. –  Ben Crowell Aug 24 '12 at 15:12
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8 Answers

up vote 17 down vote accepted

The epsilon-delta definition of a limit is often a student's first exposure to universal and existential quantifiers in a formal setting. It's important to understand each one and how "for all, there exists" differs from "there exists, for all". It was not immediately apparent to me and others in my class that these two statements are wildly different.

Getting the quantifiers straight is also important for helping students understand which variable to "fix" and which to "choose". When the definition says "there exists a $\delta$", students can confuse that as something that is given and that they somehow need to prove that this mysterious $\delta$ works for all $\epsilon$.

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Definitely! And as someone that just got done teaching this class, I have to say that presenting examples for why they are different and making the students produce examples doesn't help that much. Maybe I'm too pessimistic, but the key problem is that students want to get by with absolute minimal effort and so they don't spend any time thinking about the examples to figure out the different. They just blindly write down things and turn them in. –  Matt Aug 24 '12 at 14:13
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I think that carefully using "for each" in place of "for all" can help. I think it makes it clearer that a unique instance of this statement applies to each value of this variable. For better or worse, the $\epsilon$-$\delta$ definition is taught to a lot of people who are just taking calculus because their degree requires it. –  axblount Aug 24 '12 at 14:37
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I think a personal problem I had with epsilontics when first learning it that the proofs tend to be not very indicative of how one can arrive at them. One chooses values according to some inequality and verifies that it works out. The process of actually arriving at the inequalities involved is hidden in such proofs, but essential to creating them.

So I think it is particularly hard to learn epsilontics by simply working through a lot of epsilontics. Later one develps some intuitions and heuristics for getting to these inequaities, but its hard to explicate them. Also, this is one of the first occurances of proofs requiring a significant amount of creativity, they are not purely mechanical.

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Among the difficulties, I feel that students have trouble with the logic. When trying to construct a $\delta$ for a given $\epsilon$, it is often necessary to work backwards. You then notice that all your inequalities were if and only if. So, you can work them in the reverse, and logically valid, direction. Students tend not to understand what it means to assume what you are trying to prove.

Then, when presented with an example that cannot be worked out in the reverse direction, they try to do so. When you tell them that will not work, they often have trouble trying to figure out how to even start the problem. They lack the tricks to find the appropriate $\delta$.

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I am a student, and I always find epsilon-delta proofs hard. The reason is the notation, and the number of things to keep in your head at the same time. It is always something like:

For any epsilon, there exists a delta, such that it holds that ...

Once you get to really understand the underlying mechanics, the epsilon/delta proofs are not that hard - they are just really scary looking!

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One major roadblock that students encounter is that the epsilon-delta proof is a conditional statement, and a direct proof involves assuming the hypothesis. Many Calc I students have not seen logic (even $P \Rightarrow Q$)...

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I think the $\epsilon$-$\delta$ definition is difficult to the students who don't know where the definition comes from. I have once tried to teach a student by asking what it means for a sequence to converge and let him make up his own definition. I did give a lot of hints here and there, but I made him think through it. It was a lengthy discussion but fun. Later, the $\epsilon$-$\delta$ definition was built informally by converting "all sequences" into an interval of length $\delta$ (or $2\delta$ if you prefer). This conversion was understood pretty quickly, probably because the concept is the same.

Example situation:

I: How do you define convergence of a series?

Student: It's when the number tends toward something.

I: How do you define "tends toward something"?

Student: ... It gets closer and closer to some number.

I: Ok, so if I have this sequence: $1, 0, \frac 12, 0, \frac 13, 0, \frac 14, 0, \ldots$, does it converge?

Student: It does because it eventually goes to zero.

I: But $\frac 14$ is farther away from zero than the term before it.

Student: ...But afterwards, everything is smaller than $\frac 14$.

And the discussion goes on. These are not exact words, but the content is close to what I actually discussed.

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Personally, I find the epsilon-delta proofs start to get difficult when students have to prove that one part $X$ is less than $\epsilon /2$, another part $Y$ is less than $\epsilon /2$, so their sum $X+Y <\epsilon$.

More advanced proofs even have a $\epsilon /3+\epsilon /3 +\epsilon /3 <\epsilon$ style of proof.

Sometimes it is not so obvious to students how to get started.

One method I find useful is to think in terms of limits first, non-rigorously, and then translate the idea to epsilon-delta notation.

A pictorial visualisation would probably be of great help too, for beginners.

The problem is most lecturers (especially analysts) teaching analysis are so good at it, they don't use the pictorial visualisation or the "non-rigorous" way of thinking to get the idea.

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This practice always seems a bit silly to me. It's generally clear that attaining $X+Y < 2\epsilon$ is equivalent, since $\epsilon$ is arbitrary. –  Jair Taylor Aug 24 '12 at 21:06
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Sometimes language is the obstacle. I stumbled over the difference between 'if' and 'if and only if' when the textbook didn't provide clear enough examples. Linguistically, my math textbook was a bit loose with its explanations for what I'm sure were mathematically rigorous terms. For someone who pays attention to the language used, this can turn a small problem into a large problem when the explanation introduces ambiguity due to said linguistic laxness.

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