V. Creating L over [r1, r2] where r1, r2 are Real Numbers
Let:
S = {ℝ}
IS = [r1, r2] where r1 < r2
Partition IS to create the list L of real numbers between r1 and r2.
IS = [r1, r2] L = (r1, r2)
= [r1, [r3], r2] L = (r1, r3, r2)
= [r1, r3], [r3, r2]
= [r1, [r4], r3], [r3, [r5], r2] L = (r1, r4, r3, r5, r2)
= [r1, r4], [r4, r3], [r3, r5], [r5, r2]
= [r1, [r6], r4], [r4, [r7], r3], [r3, [r8], r5], [r5, [r9], r2] L = (r1, r6, r4, r7, r3, r8, r5, r9, r2)
= [r1, r6], [r6, r4], [r4, r7], [r7, r3], [r3, r8], [r8, r5], [r5, r9], [r9, r2]
… …
At the limit of the process, L will appear as follows: L = (r1,…r6,…r4,…r7,…r3,…r8,…r5,…r9,…r2).
By definition 13 there are no immediate predecessors or successors in [r1, r2]. It follows that the partitioning of sub-intervals of [r1, r2] can go on indefinitely. Also, except for r1 and r2, every number in the original interval must, at some point during the process, become a relative bound and only then added to L. And because no number will be a relative bound more than once, there will be no duplicates in L.
As can be seen in the examples above, each relative bound becomes the lower bound of one sub-interval and the upper bound of another sub-interval. This means that every number in the each sub-interval will be approached from left below and right its value and the interval lengths will become infinitesimally small.
Example 3, using numbers:
Let:
S = {ℝ}
IS = [1, 4]
Partition IS to create the list L of real numbers between 1 and 4.
IS = [1, 4] L = (1, 4)
= [1, [π], 4] L = (1, π, 4)
= [1, π], [π, 4]
= [1, [e], π], [π, [3.2], 4] L = (1, e, π, 3.2, 4)
= [1, e], [e, π], [π, 3.2], [3.2, 4]
= [1, [√2], e], [e, [3], π], [π, [3.15], 3.2], [3.2, [3.3], 4] L = (1, √2, e, 3, π, 3.15, 3.2, 3.3, 4)
= [1, √2], [√2, e], [e, 3], [3, π], [π, 3.15], [3.15, 3.2], [3.2, 3.3], [3.3, 4]
…
At the limit of the process L = (1,…√2,…e,…3,…π,…3.15,…3.2,…3.3,…4).