Using Archemedian property, prove that the set of natural numbers is not bounded above

To demonstrate that the set of natural numbers is not bounded above using the Archimedean property, we first need to clarify what the Archimedean property states. Essentially, it asserts that for any two positive numbers, no matter how large one of them is, there exists a natural number that can exceed the other. This property is fundamental in understanding the behavior of natural numbers in relation to other numerical sets.

Understanding the Archimedean Property

The Archimedean property can be formally stated as follows: For any real numbers \( x \) and \( y \) where \( x > 0 \) and \( y \) is any real number, there exists a natural number \( n \) such that \( n \cdot x > y \). This means that no matter how large \( y \) is, we can always find a natural number \( n \) that, when multiplied by \( x \), will surpass \( y \).

Applying the Property to Natural Numbers

Let’s apply this property to show that the set of natural numbers \( \mathbb{N} \) is not bounded above. We can choose any real number \( M \) as a potential upper bound for the natural numbers. According to the Archimedean property, we can select \( x = 1 \) (a positive number) and \( y = M \) (our upper bound).

• Since \( x = 1 \) is positive, we can find a natural number \( n \) such that \( n \cdot 1 > M \).
• This simplifies to \( n > M \).

What this means is that for any chosen upper bound \( M \), we can always find a natural number \( n \) that is greater than \( M \). Thus, no matter how large \( M \) is, there will always be a natural number that exceeds it.

Illustrative Example

Let’s consider a specific example. Suppose we take \( M = 1000 \). According to the Archimedean property, we can find a natural number \( n \) such that \( n > 1000 \). In fact, we can simply choose \( n = 1001 \), which clearly satisfies the condition. If we had chosen \( M = 1,000,000 \), we could find \( n = 1,000,001 \), and so on.

Conclusion of the Argument

Since we can always find a natural number greater than any proposed upper bound \( M \), we conclude that the set of natural numbers is not bounded above. This aligns perfectly with the Archimedean property, reinforcing our understanding of the infinite nature of natural numbers. Therefore, we can confidently assert that there is no maximum natural number, and the set extends indefinitely.