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## Nested intervals

Let be a collection of closed bounded non-empty intervals.
• We say that is nested if and only if for all , or equivalently, if and only if , for any , with and in .
For each , denote by the (finite) length of the interval .
Then the sequence of lengths is a decreasing sequence: , for all , or equivalently , for any , with and in .
Since the set is bounded below by the number zero, it has an infimum, which is non-negative.
Put .
Define the set by the formula:

Then the theorem to be proved is:
• The set is a closed non-empty interval of length .
In particular we have the corollary:
• If , then is a one-element set.
Proof of the theorem:
Write , for each , where and are real numbers with and .
Put and .
Then, when , with and in , we have: , so , so .
In particular, we have , so , as stated above.
Also we have and .
So for any and in , we have .
So each , with in is an upper bound for the set , so exists and .
So is a lower bound for , so exists and we have .

We now prove that .
• First let .
Then , so , for each .
Also , so , for each .
So for each , we have , so , so .
So .
• Now let .
Then , for each .
So , for each .
So , for each , so is an upper bound for the set .
So by definition of .
Also , for each , so is a lower bound for the set .
So by definition of .
So , so .
So .
Since we have proved both and , we have the required result: , so is a closed non-empty interval.

Finally we prove that .
• For any , we have and , by definition of and .
So we have and .
Adding these inequalities, we get .
So is a lower bound for the set .
So we have .
• Let be given.

Then is not an upper bound for the set , by definition of , so exists with .
Also is not an lower bound for the set , by definition of , so exists with .

Put .

Then , so , so , so .

Also , so , so .

So exists, such that we have both and .

Adding these inequalities gives that exists, such that:

Since , by definition of , we have:

So we have proved that for any , we have:

So , so .
So and we are done.

Next: Nested intervals and the Up: Thetheoryofnestedintervals Previous: Characterizing real intervals
George A. J. Sparling 2012-04-10