b) Let open interval An =
(
− 1
n , n
)
. Find ⋂∞
i=1 Ai and ⋃∞
i=1 Ai

Given the open intervals \( A_n = \left(-\frac{1}{n}, n\right) \), we need to find:

1. The intersection of all these intervals: \( \bigcap_{i=1}^{\infty} A_i \).
2. The union of all these intervals: \( \bigcup_{i=1}^{\infty} A_i \).

### 1. Finding \( \bigcap_{i=1}^{\infty} A_i \)

The intersection of the intervals \( A_n = \left(-\frac{1}{n}, n\right) \) involves finding the set of points that belong to all \( A_n \) as \( n \) tends to infinity.

As \( n \) increases, \( -\frac{1}{n} \) gets closer to 0 from the left, and \( n \) increases without bound. Therefore, for every point \( x \) in the intersection:

- The point must satisfy \( x > -\frac{1}{n} \) for all \( n \). In the limit as \( n \to \infty \), this means \( x \geq 0 \).
- Simultaneously, the point must satisfy \( x < n \) for all \( n \), but this is always true for finite \( x \) since \( n \) is unbounded.

Thus, the only value that can be in every \( A_n \) is \( x = 0 \).

\[
\bigcap_{i=1}^{\infty} A_i = \{0\}
\]

### 2. Finding \( \bigcup_{i=1}^{\infty} A_i \)

The union of the intervals \( A_n = \left(-\frac{1}{n}, n\right) \) involves finding the set of points that belong to at least one \( A_n \).

As \( n \) increases:
- The lower bound \( -\frac{1}{n} \) approaches 0 from the left, meaning any negative number slightly less than 0 will eventually be included in some \( A_n \).
- The upper bound \( n \) grows without bound, meaning all positive numbers will be included as well.

Thus, the union of all intervals covers all real numbers:

\[
\bigcup_{i=1}^{\infty} A_i = \mathbb{R}
\]

### Final Answers

1. \( \bigcap_{i=1}^{\infty} A_i = \{0\} \)
2. \( \bigcup_{i=1}^{\infty} A_i = \mathbb{R} \)

---

Would you like more details or have any questions? 

Here are some related questions to consider:

1. What happens if the interval was \( A_n = \left(-\frac{1}{n}, \frac{1}{n}\right) \)?
2. How does the intersection change if the intervals are closed instead of open?
3. What if the intervals were of the form \( A_n = \left(0, \frac{1}{n}\right) \)?
4. How would the results differ if \( A_n = \left(-n, \frac{1}{n}\right) \)?
5. What is the significance of the result in terms of measure theory?

**Tip:** When dealing with infinite intersections, consider how the endpoints behave as \( n \) changes to understand the limiting behavior.

Let open interval An = (-1/n, n). Find ⋂∞ i=1 Ai and ⋃∞ i=1 Ai.

Explore the infinite intersection and union of open intervals An = (-1/n, n) and learn how the intersection results in {0} and the union covers all real numbers. This problem involves key concepts from real analysis and set theory, focusing on the behavior of limits and infinite intervals.

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