Give an example of an infinite subset A of the interval (0,1) with the property that for every natural number, n, the intersections of A and (1/n,1) are finite.

We need to construct an infinite subset \( A \subset (0, 1) \) such that for every natural number \( n \), the intersection \( A \cap \left( \frac{1}{n}, 1 \right) \) is finite.

### Example of such a set \( A \):
Define the set \( A \) as follows:
\[
A = \left\{ \frac{1}{k} \mid k \in \mathbb{N}, k \geq 2 \right\}
\]
This set \( A \) consists of the reciprocals of natural numbers \( k \geq 2 \), meaning:
\[
A = \left\{ \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \dots \right\}
\]

### Verifying the properties of \( A \):
1. **Infinite Set**: \( A \) is clearly infinite, as it contains all \( \frac{1}{k} \) for \( k \geq 2 \), which is an infinite number of elements.
   
2. **Intersection with \( \left( \frac{1}{n}, 1 \right) \) is finite**: For each natural number \( n \), consider the interval \( \left( \frac{1}{n}, 1 \right) \). This interval contains all numbers between \( \frac{1}{n} \) and 1, but in \( A \), only a finite number of elements fall within this range. Specifically, the only elements of \( A \) in \( \left( \frac{1}{n}, 1 \right) \) are \( \frac{1}{n+1}, \frac{1}{n+2}, \dots \), which is clearly a finite set.

Thus, \( A \cap \left( \frac{1}{n}, 1 \right) \) is finite for each \( n \), even though \( A \) itself is infinite.

### Conclusion:
The set \( A = \left\{ \frac{1}{k} \mid k \in \mathbb{N}, k \geq 2 \right\} \) satisfies the given condition: it is an infinite subset of \( (0, 1) \), but for every natural number \( n \), the intersection \( A \cap \left( \frac{1}{n}, 1 \right) \) is finite.

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Would you like further details or have any questions?

Here are 5 related questions to expand on this concept:
1. What would happen if we allowed irrational numbers in \( A \)? How could we redefine the set?
2. Can you find another example of an infinite subset of \( (0, 1) \) satisfying the same condition?
3. How would the behavior of \( A \) change if we considered intersections with intervals of the form \( \left( 0, \frac{1}{n} \right) \)?
4. How could you generalize this concept for subsets of other intervals, say \( (0,2) \)?
5. What if instead of intervals, we required finite intersections with countable subsets of \( (0,1) \)?

**Tip**: Constructing sets with specific properties often involves balancing the size and density of the set elements within different regions of the interval.

Discover an example of an infinite subset of the interval (0,1) where, for each natural number n, the intersection with (1/n,1) is finite. This solution explores set theory, intervals, and properties of rational numbers. Suitable for undergraduate mathematics students.

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