https://file.aiquickdraw.com/mathbot/file/17267641366740vs7py91.jpg

Let's analyze and solve both of the problems from the image:

### Problem 4:
Prove that:
\[
[0, 1] = \bigcap_{n=1}^{\infty} \left( -\frac{1}{n}, 1 + \frac{1}{n} \right)
\]

#### Solution:
- First, observe that the intersection is over open intervals \(\left( -\frac{1}{n}, 1 + \frac{1}{n} \right)\), which are defined for each \(n \in \mathbb{N}\). The intervals expand as \(n\) increases.
- For \(n=1\), the interval is \(\left(-1, 2\right)\), which includes all numbers between \(-1\) and \(2\).
- For \(n=2\), the interval is \(\left(-\frac{1}{2}, \frac{3}{2}\right)\), which is smaller than the previous interval.
- As \(n\) increases, the intervals shrink and approach \([0, 1]\). More formally, the endpoints of the intervals, \(-\frac{1}{n}\) and \(1 + \frac{1}{n}\), approach 0 and 1 as \(n \to \infty\).

Thus, the intersection of all these intervals for \(n = 1\) to infinity will be the closed interval \([0, 1]\). Therefore, we have:
\[
[0, 1] = \bigcap_{n=1}^{\infty} \left( -\frac{1}{n}, 1 + \frac{1}{n} \right)
\]

### Problem 5:
Prove that if \(A\) is a family of sets and \(B \in A\), then \(B \subseteq \bigcup_{A \in A} A\).

#### Solution:
- Let \(A\) be a family of sets, and \(B \in A\).
- The union \(\bigcup_{A \in A} A\) is the set of all elements that belong to at least one of the sets in \(A\). Since \(B \in A\), all elements of \(B\) are also elements of the union.
- Formally, for every \(x \in B\), since \(B \in A\), we have \(x \in B \subseteq \bigcup_{A \in A} A\).
  
Therefore, \(B \subseteq \bigcup_{A \in A} A\).

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Do you want further details on these proofs or have any questions? Here are five related questions to deepen understanding:

1. What is the significance of using open intervals in Problem 4?
2. Can you generalize Problem 4 for any interval \([a, b]\)?
3. How would the proof in Problem 5 change if \(B \notin A\)?
4. Can you give an example of a family of sets to illustrate Problem 5?
5. What are some practical applications of intersections and unions of sets in real life?

**Tip:** In proofs involving intervals, always check how the boundaries behave as \(n\) changes to ensure the correct interval limits.

Prove that [0, 1] = ⋂_{n=1}^{∞} ( -1/n, 1 + 1/n) and prove that if A is a family of sets and B ∈ A, then B ⊆ ⋃_{A ∈ A} A.

Learn how to prove that the closed interval [0, 1] is the intersection of infinite open intervals (-1/n, 1 + 1/n) and understand set theory by proving B ⊆ ⋃_{A ∈ A} A when B is a set in the family A.

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