Extract 6.8
Context: The tutor comments on the students' (Jack and Andrew) proof for CD3.2: it was fine but they have been reproducing on paper what is only acceptable as an example of b/b writing technique. Jack asks whether the tutor's proof is a 'general method' for dealing with suprema and the tutor describes the Completeness Axiom as a 'machine, as a function, on sets in which you plug in sets and provided they are bounded above and non-empty, they have a sup'. Once the existence of sup is confirmed, he continues, they then have to identify the exact number that has the supremum properties. The tutor then stresses that CD3 is a problem sheet that needs special attention and they turn to CD3.3.
In CD3.3i Andrew, says the tutor, took for granted a breaking down of the natural numbers which had not so far been proved and parts of which are proved in CD3.3ii. Andrew says he thought he could assume this breaking down because it seemed natural to him. The tutor stesses that when they think of things like these they should be more critical: what would be the point of asking them to prove CD3.3ii if they could take it for granted in CD3.3i? The tutor then asks Jack to present his proof for CD3.3i on the b/b (fig.8a).
The Episode:
Jack presents his proof for CD3.3i on the b/b (fig8.a). The tutor interrupts him in order to ask how
x-1<n£x
comes from the Archimedean Property.
J1: It's not quite [coming from the ArchPr] because for any natural number there is a... em,... for any number there is a natural number that follows and...
The tutor agrees that 'this is what the ArchPr says'. Jack is hesitant ('So there's also... So if x is any real number then...'). The tutor points out that there is nothing wrong with what he wrote but he is 'taking two steps in one' and that 'when you are saying 'by the ArchPr' you're slightly misleading your reader or your listener'. Jack looks still hesitant and the tutor points out it's 'a little more' than the ArchPr. Andrew then says that n£x means 'that the natural numbers are not bounded by the real numbers'. The tutor nods in agreement with Andrew but is still leaning towards Jack. He then stresses:
T1: So there is a natural number greater than x-1 and then what do you do? [Silence] There are natural numbers greater than x-1.
J2: So I'm taking the least.
The tutor says he agrees. Andrew wants to know why they need the least nmber greater than x-1. The tutor points out that the least number is still bigger than x-1 and Jack says that 'you can't take any natural number bigger than that. You have to take the next one along'. He completes his writing om the b/b (fig.8a). The tutor says he thinks it is an excellent proof but points out that Jack did not need to introduce the name e, since the question provides the name fracx. He then concludes: the point in this question is not merely prove the existence of these numbers but also to manufacture them.
For CD3.3ii the tutor suggests rephrasing the long sentence in the problem sheet that has been causing problems with its complexly quantified formulation: quantifiers is, he says, a great source of difficulty at this stage (see fig.8b for the rephrasing suggested by the tutor). Jack then says this is a question about number bases and asks whether 10 has been an historically arbitrary choice of a base. The tutor then talks about the history of base-10 numbers. He then says he approves of the use of Mathematical Induction in their proof even though it was not ‘very efficient’. There is a quicker way, he suggests. 'Do you go straight to the general case?' asks Jack. The tutor suggests: note that by CD3.3i int(t/r)=t1 is a natural number. By the inductive hypothesis this can be written as in CD3.3ii. Then t=rt1+rt0 is the analysis of t, requested in the question. He then suggests talking about Accumulation Points which is a crucial concept in CD3.3ii.
Return to Appendices for Chapter 6.