Context: The tutor is demonstrating 'a more attractive way', as he says, of CD2.3 and CD2.4 and discusses with student Cornelia the use of counterexamples in CD2.5. Then they turn to CD2.6.

In CD2.6i Cornelia says she showed that infS exists but she could not prove that it is equal to ksupS. She also showed that ksupS is a lower bound for kS. Therefore what she could not prove is that ksupS is the greatest lower bound of kS. The tutor suggests 'following your nose through the definitions' (fig.7): let b> ks, where s=supS, and prove b is not a lower bound for kS, namely prove that there exists an x in kS such that x<b. So what happens if we divide b>ks by k?

C1: You've got b/k is greater than s in...

T1: No, remember k is negative.

C2: It's less than...

T2: Now follow your nose, what do you know about [b/k]?

C3: It's got to be the greatest...

T3: No, come on. SupS is the least upper bound for S. So what can you say for b/k if that's the least upper bound of S?

C4: It's hard work...

T4: Yeah, it is. Here is the least upper bound and here is a number smaller than it. What can you say about it?

C5: That must be in the set.

'No, no, not necessarily!' exclaims the tutor and explains: since S can be a 'dotty kind of set', all you can say is that b/k is not an upper bound of S. Therefore there exists an a in S with a>b/k. Cornelia then points out that multiplying through with k gives that ak<b and ak is the x we were looking for.

The tutor concludes that this is an example of what he means by 'following your nose through the definition', that is 'Apply the definition at each different stage'. In this case, he adds, among all the things one could say about b/k, the useful observation is that b/k is not an upper bound of S. He then concludes: she could have picked up the right thing to say if she had focussed on her goal to prove that b cannot be a lower bound for kS.

Return to Section 6(vii).

Return to Appendices for Chapter 6.