Exploring the Benefits of Epsilon-Delta Proofs in Mathematical Analysis

[NockWodz]

I understand the concept of Epsilon-Delta proofs, but I can't understand why we have to do them.
What's the advantage of using this proof over just showing that the limit from the function approaches from the left and right are the same?

 

[RPinPA]

If you have another way of showing it other than applying the definition of limit (the epsilon-delta statement), then that's fine.

For instance, you might be able to take advantage of some limit theorems. But those theorems were most likely originally proved with epsilon-delta arguments.

 

[NockWodz]

If you have another way of showing it other than applying the definition of limit (the epsilon-delta statement), then that's fine.

For instance, you might be able to take advantage of some limit theorems. But those theorems were most likely originally proved with epsilon-delta arguments.

Ah, sorry, I probably should have specified a bit more. I'm currently working on a project where I have to teach this method to the class, and I'm worried about someone asking why this method is being taught when we have other options. So I'm wondering specifically the advantages of Epsilon-Delta proofs. (Sorry if I'm not making a lot of sense, I might not have as strong of an understanding as I thought). Thanks!

 

[PeroK]

Ah, sorry, I probably should have specified a bit more. I'm currently working on a project where I have to teach this method to the class, and I'm worried about someone asking why this method is being taught when we have other options. So I'm wondering specifically the advantages of Epsilon-Delta proofs. (Sorry if I'm not making a lot of sense, I might not have as strong of an understanding as I thought). Thanks!

What's your alternative definition of a limit; or of continuity of a function?

 

[fresh_42]

Limits are just an abbreviation. If you resolve them by their meaning, you will end up with a epsilon-delta version of continuity, It are equivalent definitions, so it's a matter of taste or context which one fits better.

 

[NockWodz]

What's your alternative definition of a limit; or of continuity of a function?

In the past we would either just know the limit exists due to continuity of a function, and thus being able to substitute x into the function, or from substituting in numbers slightly greater than and less than the x-value, each getting subsequently smaller, until they are nearly the same number.

 

[PeroK]

In the past we would either just know the limit exists due to continuity of a function, and thus being able to substitute x into the function, or from substituting in numbers slightly greater than and less than the x-value, each getting subsequently smaller, until they are nearly the same number.

That's not a definition of anything. What do you even mean by "nearly the same number"?

 

[NockWodz]

That's not a definition of anything. What do you even mean by "nearly the same number"?

Sorry. I guess the only definition of a limit we know of is we can bring a function infinitely close to a limit by choosing x-values infinitely close to the x-value we're approaching.

By "nearly the same number" I'm referring to choosing x-values from the function approaching x from the left and the right, like 2.001 and 1.999 as we approach 2, but choose small enough values that f(x) yields (nearly) the same value.

 

[fresh_42]

Sorry. I guess the only definition of a limit we know of is we can bring a function infinitely close to a limit by choosing x-values infinitely close to the x-value we're approaching.

By "nearly the same number" I'm referring to choosing x-values from the function approaching x from the left and the right, like 2.001 and 1.999 as we approach 2, but choose small enough values that f(x) yields the same value.

... and if you formally make this rigorous you end up with $\varepsilon-\delta$.

 

[NockWodz]

... and if you formally make this rigorous you end up with $\varepsilon-\delta$.

I understand that. I think I'm just making this a lot more confusing for myself than it needs to be because I'm worried people aren't going to be able to understand the way I say it. Nevermind, and sorry.

 

[PeroK]

I understand that. I think I'm just making this a lot more confusing for myself than it needs to be because I'm worried people aren't going to be able to understand the way I say it. Nevermind, and sorry.

I think it's fair to say that unless you are clear in your own mind, then it's unlikely to be clear in the minds of your students.

 

[symbolipoint]

I understand the concept of Epsilon-Delta proofs, but I can't understand why we have to do them.
What's the advantage of using this proof over just showing that the limit from the function approaches from the left and right are the same?

What I say here is not really an answer, but I feel some explanation is needed anyway.
You need to be comfortable with and can understand these epsilon-delta proofs yourself if you want to teach them for derivatives, assuming you will teach semester 1, Calculus 1, to high school or college students, and you yourself should probably have at least the equivalent of undergraduate degree in Mathematics. Use of variables to PROVE limits of functions is part of studying Calculus. These proofs must be there because just waving numbers (values) around is not enough to PROVE. Your students might have too much trouble understanding but YOU must not have too much trouble understanding nor presenting. Maybe you will need to restudy this stuff on your own before you try to teach it. Maybe you can do your own review from similar books which your students will use.

 

[DaveE]

The ε-δ concept is worth learning itself, regardless of what the specific proof is. It is a concept fundamental to calculus. If you students don't understand it, then they won't really understand calculus, IMO; even if they can still solve homework problems.

 

[statdad]

"we can bring a function infinitely close to a limit by choosing x-values infinitely close to the x-value we're approaching.

The words I've highlighted there are weasel words -- your phrasing has no meaning since you don't specify what you mean by "infinitely close".

"By "nearly the same number" I'm referring to choosing x-values from the function approaching x from the left and the right, like 2.001 and 1.999 as we approach 2, but choose small enough values that f(x) yields the same value."

More weasel words at "but choose small enough values" -- how do you know when you're close enough? How do you know that the cutoffs you choose for "close" give the correct answer for what occurs when you're even closer?

I think the very nice behavior of the functions you (and, to be honest, everyone early in their math training) are familiar with give a false indication of the behaviors of functions in general: they're too easy to picture, so your
intuitive message about what limits are "works" for them. It won't help for more complicated functions.

A couple other points:

• this "get close" idea for limits would need a good bit of wording adjustment to deal with limits at infinity -- wording that I don't believe would be easy to motivate in a logical sense. The traditional limit definition a point is easy to modify
• For any students that go on to more advanced settings the $\varepsilon$ and $\delta$ idea
• will be crucial, whether you are dealing with limits relating to integration, functions of several variables, or more abstract notions. Postponing the introduction of the notation and the familiarity of working with it will only hamper those students' progress

By all means relate the $\varepsilon$ - $\delta$ idea to the intuitive ideas, but introduce the formal definition first.

 

[Stephen Tashi]

So I'm wondering specifically the advantages of Epsilon-Delta proofs. (Sorry if I'm not making a lot of sense, I might not have as strong of an understanding as I thought). Thanks!

One way to look at it is that epsilon-delta definitions use the concept of logical quantifiers to overcome the vagueness in intuitive concepts of limit - such as a limit involving a process that takes place in time (e.g. "as x gets closer and closer") or something that involves a series of steps "as n approaches..." or something that is "infinitely small" in magnitude but not zero (e.g. $dx$ as having such properties).

A difficulty in teaching the role of quantifiers is that the intuitive concepts came first in the history of mathematics and they've left their mark in modern terminology. For example, if you verbalize the notation "$\lim_{x \rightarrow a} f(x)$", you say the words "as $x$ approaches". However, the definition of the complete sentence $\lim_{x \rightarrow a} f(x) = L$ in the modern epsilon-delta form does not define what it means for a variable to "approach" a number.

In my opinion, before students are taught mathematics that depends on definitions involving the quantifiers "for each" and "there exists", they should be taught a little about logic and logical quantifiers using examples in everyday language like books on logic employ.

They should also should understand how Logic views mathematical definitions. A mathematical definition gives the logical equivalence between an undefined statement and a previously defined statement. (The interpretation of notation should result in a statement ( i.e. a complete sentence), so defining new notation is a special case of defining a previously undefined statement.) The definition of limit of a real valued function of a real variable deals with the undefined statement $lim_{x \rightarrow a} f(x) = L$. It says that this statement is logically equivalent to the statement "for each real number epsilon that is greater than zero, there exists a real number delta such that for each real number x, if the absolute value of x minus the real number a is greater than zero and less than delta then the absolute value of the function f evaluated at x, minus L is less than epsilon".

Definitions from the viewpoint of mathematical logic are at odds with what might be called the "liberal arts" approach to definitions. In the liberal arts approach, students are taught to break an undefined statement into parts and find an interpretation of each part - even each word - and then put these interpretations back together. This is not a reliable approach in mathematics. It will only work on mathematical statements when each part of the statement has been previously defined. Some mathematical definitions define statements consisting of several previously undefined phrases without defining the meaning of the individual phrases.

The intuitive approach to limits was sufficient to develop much of calculus. The modern epsilon-delta definition was developed only when the intuitive approach ran into trouble - cases where it was ambiguous or self-contradictory. The paradox of explaining this is that students need a certain level of understanding before they can appreciate examples where the intuitive approach has problems.

Before you teach the epsilon-delta definition of limit, I suggest you glance at a text on logic.

 

[AVBs2Systems]

Hi NockWodz

I can say that we use epsilon delta proofs to prove that a limit exists because that's literally the definition of a limit.

Hence, to prove that some objects exists or is equal to some other well defined object, the way is to prove that it matches the definition of that object.

In mathematics:

Axioms, definitions, theorems are three important classifications of the formal structure, yes?

Here is the definition of an even number:

$$ 2 k \,\,\,\,\,\ k \in \mathbb{Z} $$
To prove that 50 is an even number, you have to prove that:

$$ 50 = 2 k \,\,\,\,\,\, k = 25 \in \mathbb{Z} $$Hence it is even. Likewise the limit of a function is another object like the even numbers, and you have to prove that a function has a limit, ie, prove the definition fits.

In most real analysis texts, the definitions are developed from sequences and series then moved onto continuous and discontinous real valued functions.

So now, the definition for the limit of a sequence is roughly " A sequence a sub n is convergent to X if the limit as n goes to infinity the absolute value of the difference between a sub n and X is less than all epsilon where epsilon is greater than or equal to zero."

IIRC:
$$
\forall \epsilon \gt 0 \,\,\,\,\, \exists M \,\,\,\, \text{S.T} \,\,\,\, \forall_{n} \ge M \,\,\,\, |a_{n} - X | \lt \epsilon
$$
Thats what I recall for now being the definition of a limit for a convergent sequence, anyway to prove that any sequence converges to some limit, you have to prove that the sequence matches the definition given.

So its like that.

Of course, there will be many equivalent definitions of one mathematical object, so you can choose the most complicated definition or the simplest, and I think the epsilon delta definition is probably the simplest possible, there is some topology here which basically has a higher level definition of a limit, I think, but you can see what I am getting at.

 

[mathwonk]

a proof is just a verification of a definition. I.e. before you can have a proof you must have a definition. Then you check that the conditions in the definition hold; that is a proof. The epsilon delta definition of limit is the only rigorous definition of a limit that I myself know. Once you give an epsilon delta definition, then of course you must give an epsilon delta proof. Indeed the epsilon delta definition of limit is just a precise statement of the definition of a tangent line given by Euclid long ago, hence has a long history.

But more importantly, the epsilon delta definition of a limit is one that can actually be checked in practice. I.e. given an arbitrary epsilon, you can argue that there is always a corresponding delta, and then you have proved that your limit exists. By this method you can actually prove that functions which you think should be continuous really are continuous, such as polynomials.

e.g. suppose you want to prove that the function x^2 is continuous at x=0. you start by giving an epsilon, and you try to prove there is a delta such that whenever x is closer to 0 than delta, then x^2 is closer to 0^2 = 0 than epsilon. I.e. given e>0, you want d>0 so that |x-0|<d forces |x^2-0| < e. But whenever |e| < 1, then |e|^2 < |e|, so given e>0, all you have to do is choose d = e if e ≤ 1, or d = 1 if e >1. Then if e ≤ 1, |x-0| = |x|< d = e ≤ 1, forces |x^2-0| = |x^2| ≤ |x|< d = e. I.e. |x| < d forces |x^2|<e, as desired.
And if e > 1, then |x-0| = |x| < d = 1, again forces |x^2-0| = |x^2| < |x| < 1 ≤ e. So again |x|<d forces |x^2|<e.

I am not saying this is easy to follow, but it is possible, and it proves it cold.

So the purpose of epsilon delta definitions and proofs is that they can actually be verified. Of course you should still convince yourself that what is being checked does correspond to an intuitive idea of limit and continuity.I am tired now but if you work at it perhaps you can convince yourself that Euclid's theorem that a line L through a point p of a circle is tangent to that circle provided no other line through p can be placed that lies strictly in the space between L and the circle, is equivalent to Newton's epsilon delta limit definition. I.e. if you make this statement precise you will find the epsilon delta definition of a limit; i.e. given any line M also through p and making an angle e with L, there is a small circle of radius d around p, such that within that circle, the arc of the original circle lies between the lines L and M.

To sum up, we want to give a definition of a limit so that in a specific case, we can check whether or not the given function does or does not satisfy the definition. the epsilon delta definition satisfies that requirement.

If you have an alternate definition, then test it out by trying to show that x^2 is continuous at x=0, i.e. that the limit as x-->0, of x^2, is also 0.

 

[mathwonk]

You did not ask this, but now that I have argued for why one does epsilon delta proofs, I want to remark on how one finds such proofs, e.g. for the continuity of x^2 at x=a. Although technically speaking the definition is, given e>0, find d>0 so that whenever |x-a| < d, then one also has |x^2-a^2| < e, the easiest way to prove this is by rewriting it like this; set (x-a) = h, hence x = a+h, and then show that given e>0, there is d>0 such that whenever |h| < d, then |x^2-a^2| = |(a+h)^2 - a^2| = |2ah + h^2| < e.

I.e. this is easier because the quantity you must make small, namely |x^2-a^2|, has been expressed in terms of the quantity you get to choose, namely h = (x-a). I.e. now we have |x^2-a^2| = |(a+h)^2 - a^2| = |2ah + h^2|, and it may seem more obvious that making h small will also make |2ah + h^2| small.

I.e. given e>0, if we want to make |2ah + h^2| < e, it will suffice to make each term less than e/2, so we want

|2ah| < e/2 and also |h^2| < e/2. solving, we want |h| < e/|4a|, and also |h^2| < e/2. So take |h| < e/2, and also |h| < 1, (or if you prefer, take |h| < sqrt(e/2)), and also |h| < e/|4a|. That should do it.

Similarly, if you want to prove that x^3 is continuous at x=a, write again x = a+h, so that then |x^3 - a^3| =

|(a+h)^3 - a^3|= |3a^h + 3ah^2 + h^3|, and you can try to see how small |h| should be so that this sum is less than e.

It's tedious, but this method makes it doable.

 

[FactChecker]

It's more than learning a specific proof. Anyone who will eventually major in mathematics needs to learn the discipline of using formal definitions to do proofs. The formal definition of "continuous" is in terms of $\epsilon - \delta$, so the proofs should use that.

Of course, they also need to learn to use established theorems and lemmas rather than always doing their own $\epsilon - \delta$ proof from scratch. And they need to learn when to do each approach.

 

[WWGD]

Ultimately, as a heuristic, it becomes somthing like: we can get indefinitely-close to the output f(x) of f if the input is made close-enough to x. And, as others said, this is formalized or rigorized with a $\delta - \epsilon$ argument. This is also used to show continuity.