Find the property of the real numbers which will complete the argument below:

1. Zero is not a positive number.
2. Choose x as the smallest number.
3. The ________ property tells us that _________.
4. Therefore, between x and 0 there will be a third number, smaller than x.
5. Call this new number x and return to step 2.
6. We never find a smallest number and we never arrive at 0.

The property of the real numbers that completes the argument is the Archimedean property, which states that for any positive real number x, there exists a positive integer n such that nx > 1.

1. Zero is not a positive number.
2. Choose x as the smallest number.
3. The Archimedean property tells us that for any positive real number x, there exists a positive integer n such that nx > 1.
4. Therefore, between x and 0 there will be a third number, smaller than x.
5. Call this new number x and return to step 2.
6. We never find a smallest number and we never arrive at 0.

By the Archimedean property, there will always be a positive integer n that is greater than any chosen x. This means that we can always find a third number between x and 0 that is smaller than x, making it impossible to reach zero as the smallest number. Therefore, we never find a smallest number and we never arrive at 0.