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. $\bigcup_{i=1}^{\infty} A_i = \mathbb{R}$

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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.