Show that the nested intervals property fails for the rational numbers

[Kb13]

In the field of formal rational functions, construct a nest of closed, bounded intervals whose intersection is empty. (That is, show that the Nested Intervals property fails in this field)


I know it has to involve radical 2 but because that is the only number we know is irrational but other than that I have no idea.

 

[VeeEight]

Try 'centering' your intervals around an irrational (you should know more irrationals than that, but sqrt2 is still a good choice). For example, one interval can be [sqrt2 - 1/2, sqrt2 + 1/2]. Turn this into a nested sequence and consider what happens as the length of the intervals get very small.

 

[jbunniii]

In the field of formal rational functions, construct a nest of closed, bounded intervals whose intersection is empty. (That is, show that the Nested Intervals property fails in this field)

Is there a topology assumed for this field? If so, what is it? I'm not sure what an "interval" looks like in this field.

 

[VeeEight]

I suspected he meant the set of rational numbers (since this is a common problem and he didn't know any other irrationals so odds are he doesn't know what a topology is).

 

[jbunniii]

Good point. I would like to point out one subtlety that he might otherwise overlook, namely that [sqrt2 - 1/2, sqrt2 + 1/2] in the rational field does NOT contain the endpoints since sqrt2 +/- 1/2 is irrational. It is nonetheless a closed (AND an open) interval.