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On 03.11.2024 14:57, Mikko wrote:There are no adjacent intervals. Between any two non-overlapping intervalsOn 2024-11-03 08:38:01 +0000, WM said:Not between two adjacent intervals. Such intervals must exist because space between intervals must exist. Choose a point of this space and go in both directions, find the adjacent intervals.
Apply Cantor's enumeration of the rational numbers q_n, n = 1, 2, 3, ... Cover each q_n by the intervalNo, it would not. Between any two distinct numbers, whether rational or
ε[q_n - sqrt(2)/2^n, q_n + sqrt(2)/2^n].
Let ε --> 0.
Then all intervals together have a measure m < 2ε*sqrt(2) --> 0.
By construction there are no rational numbers outside of the intervals. Further there are never two irrational numbers without a rational number between them. This however would be the case if an irrational number existed between two intervals with irrational ends.
irrational, there are both rational and irrational numbers.
Some intervals overlaps because there are infintely many rational numbersAs long as ε > 0 the intervals overlapLet ε = 1. If all intervals overlap and there is no space "between", then the measure of the real line is less than 2*sqrt(2). Therefore not all intervals overlap.
There are no adjacent intervals. Non-existent intervals don't preventAnyway, there are real numbers that are not in any interval.That is not possible because between two adjacent intervals there is no rational number and hence no irrational number.
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