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 xn 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 |xn-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 (xn) by stage N if, for all n≥N, we have that xn 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 (xn) if there exists a natural number N such that the set A absorbs the sequence (xn) by stage N.
There are several standard terms equivalent to this notion: for example, this is what is meant by saying that the sequence (xn) 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 xn tends to x as n tends to infinity can now be expressed as follows:
Every open interval centred on x absorbs the sequence (xn).
Compare this with the equivalent ‘eventually in’ formulation:
The sequence (xn) 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 (xn) 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