# Show that the nested intervals property fails for the rational numbers

• Kb13
In summary, the conversation discusses constructing a nest of closed, bounded intervals in the field of formal rational functions to show that the Nested Intervals property fails in this field. This involves using an irrational number, such as sqrt2, as the center of the intervals and considering what happens as the intervals get smaller. The conversation also touches on the topology of the field and the subtlety of irrational numbers not being included in the intervals.

#### 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.

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.

Kb13 said:
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.

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).

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.