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I'm inclined to talk about images like shooting an arrow at a target and such, but I'm sure you've heard these. The trouble is that it is very difficult to communicate why we'd want to move from the intuitive notion of limit to the modern one. Most of the arguments we'd give simply aren't convincing for students.

This will not be a popular answer, but one way around the above difficulty is to take a formalist approach and give the students a more or less mechanical way to deal with the multiple quantifiers. While doing this, students get a sense of the proper use of quantifiers and get an idea of how to "do" the proofs. After they gain some ability, they seem more willing to see how it affords clarity. My students have responded well to the treatment of basic proof found in Chapter 3 of Daniel J. Velleman's book How to Prove It: a structured approach.

Remember, in the question, the OP asked for ways to teach students to construct basic epsilon-delta proofs.

Here's an example: Prove that $lim_{x \rightarrow 0}2=2$. The students write two columns on their page a "givens" column and a "goal" column. In the goal column write the definition of the limit in question symbolically. Then show them that when they see a universal quantifier in the goal column, they can move it to the givens column as "Let $\epsilon$>0 be arbitrary." (This temporarily gets around student misunderstanding of the use of the word 'arbitrary'.) Next, they must manufacture a delta. Here you show them where the "real mathematics" takes place: do what is needed to manufacture your delta. Start with what you are trying to estimate, and work backwards. In our example we see that any positive delta will do. The important thing is that after doing your "scratchwork", you go back and write Let $\delta$ equal 3, (or whatever you picked) in the givens column, (you may explain to them the reason this is logical, or talk about it later). The point is, they have seen you discover the delta and that the proof is written in "reverse order". One can then remove the existential quantifier from the goal column. The goal is now simply to run your scratchwork in reverse until you end up with something formally identical to what appears in the goal column.

The idea is, if students can actually construct such a few proofs like this, maybe you will have a chance to discuss what happened. If it seems too miraculous or divorced from common sense, they simply won't listen.

After some experience has been gained (I do this looking at sequence limits instead of epsilon-delta) it is nice to give some informal yet precise descriptions of what a limit is. For example, a sequence $a_{n}$ of real numbers converges to a real number $L$ when for any $\epsilon>0$ all but finitely many terms of the sequence lie within $\epsilon$ of $L$. After they know how to construct proofs, trying to get students to conceptualize seems to work better.

Best of luck with this, Frank! (The question will probably close, but if it does I'll move the answer to math.stackexchange if you move the question there.)

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I'm inclined to talk about images like shooting an arrow at a target and such, but I'm sure you've heard these. The trouble is that it is very difficult to communicate why we'd want to move from the intuitive notion of limit to the modern one. Most of the arguments we'd give simply aren't convincing for students.

This will not be a popular answer, but one way around the above difficulty is to take a formalist approach and give the students a more or less mechanical way to deal with the multiple quantifiers. While doing this, students get a sense of the proper use of quantifiers and get an idea of how to "do" the proofs. After they gain some ability, they seem more willing to see how it affords clarity. My students have responded well to the treatment of basic proof found in Chapter 3 of Daniel J. Velleman's book How to Prove It: a structured approach.

Remember, in the question, the OP asked for ways to teach students to construct basic epsilon-delta proofs.

Here's an example: Prove that $lim_{x \rightarrow 0}2=2$. The students write two columns on their page a "givens" column and a "goal" column. In the goal column write the definition of the limit in question symbolically. Then show them that when they see a universal quantifier in the goal column, they can move it to the givens column as "Let $\epsilon$>0 be arbitrary." (This temporarily gets around student misunderstanding of the use of the word 'arbitrary'.) Next, they must manufacture a delta. Here you show them where the "real mathematics" takes place: do what is needed to manufacture your delta. Start with what you are trying to estimate, and work backwards. In our example we see that any positive delta will do. The important thing is that after doing your "scratchwork", you go back and write Let $\delta$ equal 3, (or whatever you picked) in the givens column, (you may explain to them the reason this is logical, or talk about it later). The point is, they have seen you discover the delta and that the proof is written in "reverse order". One can then remove the existential quantifier from the goal column. The goal is now simply to run your scratchwork in reverse until you end up with something formally identical to what appears in the goal column.

The idea is, if students can actually construct such a few proofs like this, maybe you will have a chance to discuss what happened. If it seems too miraculous or divorced from common sense, they simply won't listen.

After some experience has been gained (I do this looking at sequence limits instead of epsilon-delta) it is nice to give some informal yet precise descriptions of what a limit is. For example, a sequence $a_{n}$ of real numbers converges to a real number $L$ when for any $\epsilon>0$ all but finitely many terms of the sequence lie within $\epsilon$ of $L$. After they know how to construct proofs, trying to get students to conceptualize seems to work better.

Best of luck with this, Frank! (The question will probably close, but if it does I'll move the answer to math.stackexchange if you move the question there.)

I'm inclined to talk about images like shooting an arrow at a target and such, but I'm sure you've heard these. The trouble is that it is very difficult to communicate why we'd want to move from the intuitive notion of limit to the modern one. Most of the arguments we'd give simply aren't convincing for students.

This will not be a popular answer, but one way around the above difficulty is to take a formalist approach and give the students a more or less mechanical way to deal with the multiple quantifiers. While doing this, students get a sense of the proper use of quantifiers and get an idea of how to "do" the proofs. After they gain some ability, they seem more willing to see how it affords clarity. My students have responded well to the treatment of basic proof found in Chapter 3 of Daniel J. Velleman's book How to Prove It: a structured approach.

Remember, in the question, the OP asked for ways to teach students to construct basic epsilon-delta proofs.

Here's an example: Prove that $lim_{x \rightarrow 0}2=2$. The students write two columns on their page a "givens" column and a "goal" column. In the goal column write the definition of the limit in question symbolically. Then show them that when they see a universal quantifier in the goal column, they can move it to the givens column as "Let $\epsilon$>0 be arbitrary." (This temporarily gets around student misunderstanding of the use of the word 'arbitrary'.) Next, they must manufacture a delta. Here you show them where the "real mathematics" takes place: do what is needed to manufacture your delta. Start with what you are trying to estimate, and work backwards. In our example we see that any positive delta will do. The important thing is that after doing your "scratchwork", you go back and write Let $\delta$ equal 3, (or whatever you picked) in the givens column, (you may explain to them the reason this is logical, or talk about it later). The point is, they have seen you discover the delta and that the proof is written in "reverse order". One can then remove the existential quantifier from the goal column. The goal is now simply to run your scratchwork in reverse until you end up with something formally identical to what appears in the goal column.

The idea is, if students can actually construct such a few proofs like this, maybe you will have a chance to discuss what happened. If it seems miraculous, they simply won't listen.

After some experience has been gained (I do this looking at sequence limits instead of epsilon-delta) it is nice to give some informal yet precise descriptions of what a limit is. For example, a sequence $a_{n}$ of real numbers converges to a real number $L$ when for any $\epsilon>0$ all but finitely many terms of the sequence lie within $\epsilon$ of $L$. After they know how to construct proofs, trying to get students to conceptualize seems to work better.

Best of luck with this, Frank! (The question will probably close, but if it does I'll move the answer to math.stackexchange if you move the question there.)