Prove the following conjecture: Between any two unique Real Numbers, there exists another Real Number.

pick two real numbers, m, and n, with n >m

the mean of the n and m is (n+m)/2
but (n+m)/2 is a rational real number, and the mean of any two numbers is between the numbers.
QED

To prove the conjecture that between any two unique real numbers there exists another real number, we can use the completeness property of the real numbers.

The completeness property states that the real numbers are complete, meaning that every non-empty set of real numbers that is bounded above has a least upper bound (supremum) in the real numbers.

Let's assume we have two unique real numbers, A and B, where A < B.

Case 1: A and B are consecutive integers
If A and B are consecutive integers, we can prove that there exists another real number between them. Since there are infinite integers between two consecutive integers, we can simply take the average of A and B, which will be a real number.

Case 2: A and B are not consecutive integers
If A and B are not consecutive integers, we can use the completeness property of the real numbers to prove that there exists another real number between them.

Consider the set S = {x | x is a real number such that A < x < B}. This set is non-empty since B is greater than A. According to the completeness property, S must have a least upper bound (supremum) in the real numbers. Let's call this supremum C.

To prove that C is a real number between A and B, we need to show that A < C < B.

Since C is the least upper bound of S, it means that C is an upper bound for S, i.e., C >= x for all x in S. If C = B, then B would be an upper bound for S, and we can find a smaller upper bound for S. This contradicts the definition of C as the least upper bound. Therefore, C cannot be equal to B.

Similarly, if C < B, then there exists a number greater than C that is also less than B, contradicting the definition of C as the least upper bound. Therefore, C cannot be less than B.

Hence, we can conclude that A < C < B, which proves that there exists another real number C between A and B.

In both cases, we have proved that between any two unique real numbers, there exists another real number.