The Completeness Property, specifically in the context of real numbers, states that every non-empty set of real numbers that is bounded above has a least upper bound (or supremum) that is also a real number.

This property does not hold for the rational numbers ($\mathbb{Q}$) due to the presence of "gaps" in $\mathbb{Q}$. Here’s why:

Explanation:

1. Definition of Supremum: The supremum (least upper bound) of a set $S$ is the smallest real number that is greater than or equal to every element of $S$.

2. Example of a Bounded Set in $\mathbb{Q}$: Consider the set $S = \{q \in \mathbb{Q} \mid q^2 < 2\}$. This is a set of all rational numbers whose squares are less than 2.

3. Absence of Supremum in $\mathbb{Q}$:

   • The set $S$ is bounded above in $\mathbb{Q}$ because, for example, 2 is an upper bound (since $1.5^2 = 2.25$).
   • However, the least upper bound (supremum) of $S$ in $\mathbb{R}$ is $\sqrt{2}$.
   • Since $\sqrt{2}$ is irrational, it does not belong to $\mathbb{Q}$.

4. Conclusion: The set $S$ has no least upper bound in $\mathbb{Q}$ because any rational number you choose will either be less than or greater than $\sqrt{2}$, but never exactly equal to it. Thus, the supremum of $S$ does not exist in the rational numbers.

This example illustrates that not every non-empty set of rational numbers that is bounded above has a least upper bound in $\mathbb{Q}$, hence the Completeness Property does not hold.

Would you like more details or have any questions?

Here are 5 related questions:

1. What is the difference between rational numbers and real numbers?
2. How do irrational numbers fill the "gaps" in rational numbers?
3. Can you provide another example where the supremum does not exist in $\mathbb{Q}$?
4. How does the Completeness Property of real numbers relate to the concept of Dedekind cuts?
5. What are the implications of the Completeness Property for calculus?

Tip: Understanding the properties of real numbers is crucial for studying calculus and analysis, as many fundamental theorems rely on the completeness of $\mathbb{R}$.