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.

---

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.