There is no such a thing "the real number interval".
Real number is not an interval.
Given English is not your first language one might presume that you have simply misunderstood the statement. However since we have spent a considerable amount of time discussing intervals particularly those involving real numbers it appears only an attempt at obfuscation by you.
Real number is the element that enables the existence of an interval, in the first place.
For example [X,X] existence depends on X existence, and not vice versa.
Technically it is ordering that enables the existence of an interval or more specifically members in a set representing and interval.
Since this is the case then any "<" relation between intervals, is actually the "<" relation between the numbers of the intervals.
You continue to claim that yet still have not shown your work comparing each element in the two intervals you claim to have been “done” comparing before.
Furthermore, suppose that an empty interval [,] is a valid interval, but since it has no numbers, we cannot use "<" relation in its case.
Big deal you are using something that is not an interval to make a point that is not about intervals.
You are confused ordering is a specific requirement of an interval not the result or something ‘used’ as the result of one. The interval [A,B] results in an empty set when B < A. So the ordering relations “<” and “>”, or more specifically “≤” and “≥” when boundaries are included, are an intrinsic part of an interval as is the fact that there must be an A and a B (even if A = B), indicative of the ordering in that interval notation. [A] is not an interval but [A,A] is
If A = B then [A,B] or [B,A] results in just A which is the same as B
If A < B then [A,B] results in a non-empty set while [B,A] results in an empty set
If A > B then [A,B] results in a empty set while [B,A] results in an non-empty set
In other words, "<" has a meaning only between numbers (in the case of intervals) and since we deal here with real numbers, and no real number has an immediate successor, then so is the case with [1,2) interval and number2; 2 is not an immediate successor of any real number of that interval, whether you like it or not.
Since ordering is an intrinsic part of an interval ordering relations do indeed have meaning when applied to intervals or between intervals and numbers, without that intrinsic meaning the interval itself would be meaningless.
Again
Please identify the real number between the real number interval (or interval in the reals if you insist) [1,2) and the real number 2 or any real number in that interval that is not less then 2.
