can you do it without the archimedean property of reals

Certainly! We can prove the existence and uniqueness of an integer \( n \) such that \( n \leq r < n + 1 \) for a rational number \( r \) without relying on the Archimedean property.

### Proof:

#### Step 1: Existence
Let \( r \) be a rational number, so \( r \) can be written as \( r = \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b > 0 \).

We need to show that there exists an integer \( n \) such that \( n \leq r < n + 1 \). To find such an \( n \), we first multiply the inequality by \( b \) (since \( b > 0 \)) to eliminate the fraction:

\[
b \cdot n \leq a < b \cdot (n + 1)
\]

This simplifies to:

\[
bn \leq a < bn + b
\]

Now, let's define \( n \) as:

\[
n = \left\lfloor \frac{a}{b} \right\rfloor
\]

where \( \left\lfloor \frac{a}{b} \right\rfloor \) is the floor of \( \frac{a}{b} \), which is the largest integer less than or equal to \( \frac{a}{b} \).

By definition, we have:

\[
n \leq \frac{a}{b} < n + 1
\]

This implies:

\[
n \leq r < n + 1
\]

Thus, we have found an integer \( n \) that satisfies the inequality.

#### Step 2: Uniqueness
Suppose there exist two different integers \( n_1 \) and \( n_2 \) such that both satisfy \( n_1 \leq r < n_1 + 1 \) and \( n_2 \leq r < n_2 + 1 \). Without loss of generality, assume \( n_1 < n_2 \).

Then, we would have:

\[
n_1 < n_2 \leq r < n_1 + 1
\]

But this implies:

\[
n_1 + 1 > n_2
\]

Since \( n_2 \) is an integer and \( n_2 > n_1 \), we must have \( n_2 = n_1 + 1 \). However, this leads to a contradiction because \( r \) cannot be both \( \geq n_2 \) and \( < n_1 + 1 = n_2 \).

Therefore, \( n_1 \) must be equal to \( n_2 \), proving that \( n \) is unique.

### Conclusion:
For any rational number \( r = \frac{a}{b} \), there exists a unique integer \( n \) such that \( n \leq r < n + 1 \), without relying on the Archimedean property of real numbers.

Would you like any further explanations or have any questions? Here are some related questions you might consider:

1. How does the proof change if \( r \) is an irrational number?
2. What is the significance of the floor function in number theory?
3. How can this concept be applied to solve problems in modular arithmetic?
4. Is there a similar approach to finding \( n \) for complex numbers?
5. How does this proof relate to the concept of divisibility?

**Tip:** When proving mathematical properties, it's often useful to try different approaches, like avoiding reliance on more abstract concepts, to deepen your understanding.

Explore a proof showing the existence and uniqueness of an integer n such that n ≤ r < n + 1 for a rational number r. This mathematical proof does not rely on the Archimedean property of real numbers. Suitable for advanced high school students.

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