Advantages of the sequence definition of limits - MathOverflow

Advantages of the sequence definition of limits

2012-08-30T09:37:53Z

I will be teaching an introductory analysis course in the coming semester. In it the students will learn about limits of real sequences, and then will learn about limits of functions in terms of sequences.

More precisely, we will say that $\lim_{x\to a+}f(x) = L$ if whenever $(x_n)$ converges to $a$ with $x_n>a$ for all $n$, we have $\lim_{n\to \infty} f(x_n) = L$. Likewise, we will say that $\lim_{x\to a-}f(x) = L$ if whenever $(x_n)$ converges to $a$ with $x_n< a$ for all $n$, we have $\lim_{n\to \infty} f(x_n) = L$. Then we say that $\lim_{x\to a} f(x) = L$ if both $\lim_{x\to a+}f(x) = L$ and $\lim_{x\to a-}f(x) = L$.

The students will have already seen the $\varepsilon$-$\delta$ definitions of limits of functions in their calculus course. The question then is, how to properly motivate this second (equivalent) definition of limits of functions?

Are there any arguments which become significantly easier when using the sequence definition of limits of functions in place of the $\varepsilon$-$\delta$ definition? (These should be elementary enough to be understood by first year Mathematics undergraduates.)

For instance, I suppose that once one has the Algebra of Limits for sequences, one gets the Algebra of Limits for functions for free. But I'm not convinced that much is to be gained from doing things this way around.

Edit: Thanks for the answers and comments so far. It seems many people are in favour of teaching the sequence definition of limits alongside the $\varepsilon$-$\delta$ definition. I agree that it should be useful to be aware of both definitions. To be certain of this, however, I would still like to see an example of a proof which is simpler when using the sequence definition.

Answer by Liviu Nicolaescu for Advantages of the sequence definition of limits

2012-08-30T10:16:44Z

I can only speak of my experience. I was first taught convergence of sequences and then later on, the definition of limits with $\varepsilon-\delta$ and its equivalent form involving sequences. I still find this most intuitive, but then this opinion is clearly heavily influenced by my upbringing.

For disclosure, I received my math education behind the Iron Curtain, and this model was employed by most (now former) communist countries. During that time curricula in most of those countries were devised by influential mathematicians who could also communicate math very well. (For example, in USSR Kolmogorov was deeply involved in shaping math education. He even wrote some high-school textbooks that were widely used.)

What I am attempting to communicate here is that the sequences-first system was adopted by informed mathematicians who cared about math education, it was tested on a large scale (tens if not hundreds of millions of students) for a long time (several decades). Arguably this system has produced good results.

Terry Tao's textbook on analysis (which I like very much for several reasons) also relies on a sequences-first approach.

Answer by fedja for Advantages of the sequence definition of limits

2012-08-30T11:43:13Z

Interestingly enough, as far as I remember, we did limits of sequences first (without functions) and then limits of functions (with $\varepsilon-\delta$) and then, as a remark, the connection mentioned above behind the very same Iron Curtain. Actually, in the end of our "limits course", our professor either did limits over nets, or stopped one step short of it: he certainly said all necessary words and made it clear that to talk about of a limit of a mapping, you just need some set of "catchers" in the range space and some set of "tails" in the argument space. That was a bit tough in the beginning but paid off nicely when doing Riemann integration where the tails are either partitions of small mesh or partitions subordinated to a fixed partition. I still find this abstract view rather enlightening; much more enlightening that the lemma in question, which, IMHO, only makes the concept more confusing (though is quite useful as a technical tool). The main reason for this opinion is that this abstract view is unifying: all notions of limit that the students will ever meet fall under this idea, only the choices of catchers and tails vary and only one magic phrase is ever needed: "For every catcher, there is a tail whose image is contained in the catcher". The lemma you mentioned is separating: if used as a definition rather than a remark, it creates an impression that there are many ad hoc concepts of limits that all have to be understood and memorized separately, creating quite a mess in the student's head. The $\varepsilon-\delta$ definition is already hard because it mixes the limit concept and the technical descriptions of the catchers and the tails, i.e., 3 things that can be easily separated and on which you can train the students one by one if you start with the abstract view. To be honest, I haven't tried it myself in the USA yet but I certainly will when teaching freshman analysis (so far it was either business calculus, where the game is never worth the candles, or advanced courses where the concept of limit was assumed to be well-known already).

Answer by Christian Blatter for Advantages of the sequence definition of limits

2012-08-30T20:36:37Z

In my view there are no advantages in defining function limits via sequences, and this practice should be abolished. Using this definition you would have to test $\aleph_1^{\aleph_0}$ or so sequences to prove a single instance of $\lim_{x\to a} f(x)=\alpha$. Why should one bring all these sequences into the picture?

The idea of "limit of $f(x)$ when $x\to a$" is the answer to the following question: What is the "natural" value of $f$ at the special, maybe "ideal", point $a\ $? Well, it's the value that would make $f$ continuous there.

This brings me to the main point: The primary and sufficiently intuitive notion is the notion of continuity. Unfortunately the simple concept of Lipschitz continuity does not cover all cases we'd like to handle, e.g. $\sqrt{|x|}$ at $x=0$. Therefore we have to dig deeper and come up with $\epsilon$ and $\delta$, and on, and on.

Sequences, on the other hand, are a fundamental tool to construct new objects, like $e$, or $\sqrt{2}$. Of course in passing we would then prove that $\lim_{x\to a}f(x)=\alpha$ iff for all sequences $\ldots$

Answer by Ronnie Brown for Advantages of the sequence definition of limits

2012-08-30T22:30:11Z

In teaching a second year UK course in analysis some years ago I was surprised to find out how pleasant were the proofs of the some of the basic results using sequential methods, for example that a continuous function on a closed bounded subset of $\mathbb R^n$ to $\mathbb R$ is bounded. This inspired me to work out a paper on the notion of a $1$-point sequential compactification: add another point and let the sequences in $X$ with no convergent subsequence converge to the extra point! It got published too in the JLMS.

However for continuity of a function I do like to rely on the neighbourhood definition since a neighbourhood is a geometric object one can draw, whereas the $\epsilon - \delta$ are only measurements of the sizes of neighbourhoods.

Answer by Amir Asghari for Advantages of the sequence definition of limits

2013-05-05T20:06:54Z

"Are there any arguments which become significantly easier...". There is an obvious choice: Disprove that a certain function has a limit at a certain point. Showing that with epsilon-delta would be a mission impossible for most students. Using the sequence definition makes it much easier and more appealing.