Quantifier packaging when teaching convergence of sequences

Here is another one of my old posts, from way back in 2008. In fact I originally posted this here on Blogger, before I was persuaded to move over to WordPress. (In those days LaTeX wasn't available on Blogger. But now I can use MathJax here.) I deleted the version here on Blogger at some point, but now I have come full circle!

I still think that the notion of absorption (and similar "quantifier packaging" ideas) could be very helpful to at least some students! But, as far as I know, the notion hasn't really really caught on, and I only have anecdotal evidence supporting my approach.

See the WordPress post for comments on this idea from, among others, Professor Sir Tim Gowers.

I'm repeating the post here mostly because the maths looks better here now using MathJax (though this particular example isn't a fair comparison, since the version of this post on WordPress doesn't actually use LaTeX). But also, it would be great if someone else tried some ideas like this out!

I still haven't (directly) taught convergence to our first-year students. However, the notion of absorption does appear on my optional sheet for first-years, More practice with definitions, proofs and examples

Original post follows (but using MathJax, and with a few tweaks)

It is well known that maths students find statements with multiple quantifiers difficult to break down into digestible portions in order to understand the whole statement. One example of this is the definition of convergence for a sequence of real numbers (and, later, sequences in metric spaces). Given a real number \(x\) and a sequence of real numbers \((x_n)\), the definition of the statement "\(x_n \to x\) as \(n\to \infty\)" has three quantifiers:

For all \(\varepsilon>0\), there exists a natural number \(N\) such that for all natural numbers \(n\geq N\), we have \(|x_n-x| < \varepsilon\,\)

or, using symbols, and following the convention that \(n\) is an integer by default in statements like this, \[\forall \varepsilon>0,\,\exists N \in \mathbb{N}:\,\forall n\geq N,\,|x_n-x| < \varepsilon\,.\] I am developing my own approach to breaking down this statement into digestible pieces. First I have introduced into my teaching the notion of absorption of a sequence by a set.

For a natural number \(N\), I say that a set \(A\) absorbs the sequence \((x_n)\) by stage \(N\) if, for all \(n\geq N\), we have that \(x_n\in A\).

This is a single-quantifier statement which can readily be checked by students for specific sets and sequences, and (I claim) is relatively easy to understand.

I then say that the set \(A\) absorbs the sequence \((x_n)\) if there exists a natural number \(N\) such that the set \(A\) absorbs the sequence \((x_n)\) by stage \(N\).

There are actually several standard terms equivalent to this notion: for example, this is what is meant by saying that the sequence \((x_n)\) is eventually in the set \(A\), or that all but finitely many of the terms of the sequence are in \(A\), etc. As we will see, one advantage of absorption is grammatical: it makes the set the subject of the sentence, and the sequence the object.

In terms of absorption, the definition of the statement \(x_n \to x\) as \(n\to\infty\) can now be expressed as follows:

Every open interval centred on \(x\) absorbs the sequence \((x_n)\).

[Here I intend open intervals of the form \((x-r,x+r)\) for some positive real number \(r\). These correspond to open balls in more general metric spaces. The conventions involved would be explained elsewhere.]

Compare this with the equivalent 'eventually in' formulation:

The sequence \(x_n\) is eventually in every open interval centred on \(x\).

This latter formulation appears ambiguous, and can cause problems for the students. It can clearly be made unambiguous, but only at the expense of making it somewhat unwieldy:

For all \(\varepsilon>0\), the sequence \((x_n)\) is eventually in the open interval \((x-\varepsilon,x+\varepsilon)\).

I have not yet had the opportunity to teach convergence of sequences to first year undergraduates. However, I have used the notion of absorption in teaching second-year mathematical analysis. In particular, I have used this method to teach the difference between uniform convergence and pointwise convergence for sequences of functions. This material (pdf file + audio podcast) is available from u-Now, or directly from http://unow.nottingham.ac.uk/resources/resource.aspx?hid=e29ada63-e1d3-7898-9afd-42692accd0be