Analysis question- a bit difficult!

For each integer n>=0, let [a_n, b_n] be a closed, bounded interval in R (real numbers) such that a_n <= b_n (a_n is greater than equal to b_n). Suppose that the resulting collection of intervals is pairwise disjoint, that is if n =/ m (n is not equal to m), then the intersection of [a_n, b_n] and [a_m, b_m] is empty. Prove that these intervals do not cover all of R (real numbers).

Thanks! Any help will be appreciated.

Re: Analysis question- a bit difficult!

If the interval are all disjoint then by density of the real numbers there must be real numbers in between the boundaries of the intervals. These numbers are not covered by your intervals and thus your intervals do not cover R.

The only subtlety is some sort of infinite sequence of intervals in the gaps but since these intervals are closed there are either going to be further gaps or they are going to share a boundary point which is disallowed. Not true if the intervals are open, but they're not.

You can do the work or turning this into a proof.

Re: Analysis question- a bit difficult!

Yes I believe this will do. But didn't understand the point of infinite sequence of intervals in the gaps. Can you please shed more light on that please?

Re: Analysis question- a bit difficult!

Suppose you've got some set of closed intervals you've defined already. They are disjoint. So for any two adjacent ones there is some space between the endpoint of the first and the start point of the 2nd. No matter how close you make those two points if they aren't equal there are going to be infinitely many reals between them.

Ok, so you can imagine trying to come up with some sort of infinite sequence of sub-intervals to try and cover all the reals in that gap. If you could use open intervals you could do this while keeping them all disjoint. But since your intervals all have to be closed either they are disjoint, and there are infinitely many reals between the end and starting points, or they share that same end and start point and are no longer disjoint.

Re: Analysis question- a bit difficult!

Oh I see. So the point is just that there could be infinite number of such closed and bounded intervals in the gap, however, since all of these intervals are disjoint, there is a possibility for an infinite number of gaps as well. And as such there will be infinite real numbers which will remain uncovered by the collection of such intervals. Thanks for such elaborate explanation.

Re: Analysis question- a bit difficult!

what you said is true but also there are an infinite number of reals in the smallest gap that won't be covered. So all you need is one gap.

Re: Analysis question- a bit difficult!