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.
You're wasting time attaching my name to a post.
To complete the argument, we need to fill in the blank spaces in step 3 with the property of real numbers that supports the statement being made.
The complete argument is as follows:
1. Zero is not a positive number.
2. Choose x as the smallest number.
3. The "density" property tells us that between any two real numbers, there is always another real number.
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 we need to fill in the blank is the "density" property of real numbers. The density property states that between any two real numbers, there is always another real number. This property guarantees that there are no "gaps" in the real number line, allowing us to always find a number that is smaller or larger than any given number.
In the argument, step 4 relies on the density property to assert that between x and 0, there will always be a third number smaller than x. This third number becomes the new choice for x, and the process repeats indefinitely, without reaching a smallest number or arriving at 0.