VII. Derivation of f : ℕ → [r1, r2]
We will now demonstrate that there exists a bijective function from ℕ to [r1, r2], f : ℕ → [r1, r2].
We have used the Interval Sieve Algorithm to create: L = (r1,…r6,…r4,…r7,…r3,…r8,…r5,…r9,…r2)
and have proved L is complete. It is readily apparent that for every r in the list there is an associated natural number subscript. Since L is complete, containing all numbers in [r1, r2] and each number in L is associated with a single unique natural number we can assert that f : ℕ → [r1, r2] exists.
The existence of f : ℕ → [r1, r2] confirms a one to one correspondence between the natural numbers and any closed interval of real numbers.