Quantifier packaging when teaching convergence of sequences

December 12, 2008September 5, 2021Joel

See also an updated version of this post on Blogger at https://explaining-maths.blogspot.com/2021/09/quantifier-packaging-when-teaching.html

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). The definition of the statement x_n tends to x as n tends to infinity has three quantifiers:

For all ε > 0 there exists a natural number N such that for all natural numbers n≥N we have |x_n-x| < ε

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≥N, we have that x_n is an element of A.

This is a single-quantifier statement which can readily be checked by students for specific sets and sequences.

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 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 tends to x as n tends to infinity can now be expressed as follows:

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

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 ε>0, the sequence (x_n) is eventually in the open interval (x-ε,x+ε).

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

Joel December 12, 2008 / 6:49 pm

Tim Gowers was kind enough to leave some comments on the blogger edition of this post, which I will now reproduce here.

gowers said…

Dear Joel,

I like your “absorbs” idea. In particular, it hadn’t occurred to me that one could get a grammatical advantage by focusing on the set rather than on the sequence, and I think it could be very useful indeed.

Like all teaching mathematicians, I have faced the problem of dealing with quantifiers. One recent example was when I gave a talk to a non-mathematical (or rather, only partly mathematical) audience and wanted to explain Szemer\’edi’s theorem. For that I used a different trick. Obviously I couldn’t say

$\forall \delta>0\ \forall k\in\mathbb{N}\ \exists N\in\mathbb{N}\ \forall A\subset\{1,2,\dots,N\}\ |A|\geq\delta N \Rightarrow A$ contains an arithmetic progression of length $k$ . So instead I dropped down a couple of levels of quantification by saying, “If $N$ is large enough, then every subset of $\{1,2,\dots,N\}$ of size at least $N/120$ contains an arithmetic progression of length $34$ .” I then followed that up by saying that I could have chosen any other pair of numbers instead of $120$ and $34$ .

I suppose I was using two tricks: choose specific numbers instead of arbitrary ones, and say “if $N$ is large enough” instead of “there exists an $N$ such that” . The second trick isn’t removing a quantifier but it sort of disguises it somehow.

Best wishes,

Tim

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Joel December 12, 2008 / 6:58 pm

Tim Gowers’s second comment (working within the limitations of blogger) was the following

gowers said…

Actually, I did once have a similar pedagogical idea myself but never got round to trying it out. It was to have a sequence of definitions with ever-increasing numbers of quantifiers. But the trick would be that you’d only actually see one quantifier at a time, the remaining ones being hidden in a definition that you had become used to.

For example, suppose you wanted to define the notion of a Cauchy sequence. You would present a sequence of definitions as follows.

1. A sequence has diameter at most c if no two of its terms differ by more than c.

2. A sequence has diameter at most c after N if |a_p-a_q| is at most c whenever p and q are greater than N.

3. A sequence has essential diameter at most c if there is some N for which it has diameter at most c after N. (One could perhaps say “if it eventually has diameter at most c”.)

4. A sequence is Cauchy if for every positive c it has essential diameter at most c.

The advantage of doing this is that one could ask students to do exercises on the intermediate definitions and not progress to the full definition until they were comfortable with them. For instance, one could ask about the essential diameter of the sequence 0, 3, 0, 2, 0, 1.5, 0, 1.25, 0, 1.125,… and get them to see that if c is greater than 1 then it has essential diameter at most c, but not otherwise.

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Joel December 12, 2008 / 7:01 pm

I agree with Tim’s second comment 100%, especially for students in the early/middle stages of the course. Doing things this way would give the students a much better understanding of the concepts involved.

Unfortunately, we usually find ourselves with limited time and resources to cover a significant syllabus. If we spend too much extra time on one part of the syllabus, the rest of the syllabus will suffer. So, like everything in life, it comes down to finding an acceptable balance in an imperfect world where there are not enough hours in the day!

In the latter stages of the course, I think that students need to develop the ability to package quantifiers for themselves. If we do too much of this for them, it might not be good for them!

On the specific issue of Cauchy sequences, it may be possible to argue that there is a ‘redundant’ quantifier anyway. Why do we insist on looking at all $p$ and $q$ after $N$ and consider $|a_p - a_q|$ ? Why not simply look at all $p$ after $N$ and consider $|a_p-a_N|$ ? This would lead to an equivalent definition with one less quantifier, but would perhaps disguise the true nature of the Cauchy condition?

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gowers December 13, 2008 / 5:01 pm

A tiny further comment — it occurred to me that it was pointless to use the phrase “essential diameter” when “eventual diameter” sounds almost the same and would be much more intuitively tied to what it is supposed to mean.

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JamesCrook December 15, 2008 / 11:21 pm

These are excellent ideas, but how about tackling the issue of familiarity with quantifiers head on too?

How about asking or showing the students how to express the idea of ‘checkmate’ using $\forall$ and $\exists$ notation? And then what is checkmate in 1, in 2, in 3 moves? We get alternating sequences of $\forall$ and $\exists$ as long as you like from this, but in a familiar territory. You can show De Morgan’s laws for quantifiers, show that applying them is looking at the game from the other player’s point of view. You can get across clearly that $\forall$ and $\exists$ do not commute. It is, or should be, familiar in this context.

Ask them to prove or disprove

$\forall x \in \mathbb{N}~ \exists y \in \mathbb{N}$ s.t. $y> x$ .

What about

$\exists y \in \mathbb{N}$ s.t. $\forall x \in \mathbb{N}~ y>x\,?$

Then prove or disprove

$\forall v \in \mathbb{N}~\exists z \in \mathbb{N}$ with $z>v+3$ s.t. $\forall w \in \mathbb{N}~ \exists y \in \mathbb{N}$ with $y>w+1$ s.t. $\forall x \in \mathbb{N}~ v>w>x>y>z \Rightarrow x-v=z-x$ .

Possibly the big hurdle is for students to see that formal statements using universal and existential quantifiers can be manipulated as equations – and that like all equations there are rules to what you can and can’t do. There’s a tendency for students to see the quantifiers as somehow outside of that and to treat them informally, having never thought about the rules. That is where I think their common mistakes come from.

I really like the way you both are reducing the depth of the expressions by making the inner expressions familiar ideas first.

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Joel November 17, 2009 / 7:55 pm

You can now see a screencast of me discussing sequence convergence and absorption in a workshop for my second-year mathematical analysis students.
This screencast is available at
http://wirksworthii.nottingham.ac.uk/webcast/maths/G12MAN-09-10/EC4b/

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