https://r2.erweima.ai/mathbot/image/17263320941204fmrd9c7.jpg

Here are the three problems from the image:

1. **Prove**: A subset \( B \) of \( \mathbb{R}^n \) is closed if and only if \( d(p, B) = 0 \) implies \( p \in B \), where \( d(p, B) = \inf \{ d(p, q) : q \in B \} \).

2. **Prove**: \( A \cup A' \) is closed for any set \( A \), where \( A' \) denotes the set of limit points of \( A \).

3. **Prove**: \( A \cup A' \) is the smallest closed set containing \( A \), i.e., if \( F \) is closed and \( A \subseteq F \subseteq A \cup A' \), then \( F = A \cup A' \).

Let me go ahead and outline the solutions for these problems.

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### 1. Prove: \( B \subseteq \mathbb{R}^n \) is closed if and only if \( d(p, B) = 0 \) implies \( p \in B \).

- **Closed sets**: A set \( B \) is closed if it contains all of its limit points, i.e., if any sequence \( (p_n) \in B \) that converges to a point \( p \in \mathbb{R}^n \) has \( p \in B \).

- **Distance function**: The distance from a point \( p \in \mathbb{R}^n \) to the set \( B \) is defined as:
  \[
  d(p, B) = \inf \{ d(p, q) : q \in B \}.
  \]
  If \( p \in B \), then clearly \( d(p, B) = 0 \). The converse is also true: if \( d(p, B) = 0 \), then for every \( \epsilon > 0 \), there exists a point \( q \in B \) such that \( d(p, q) < \epsilon \). This implies that \( p \) is a limit point of \( B \), and if \( B \) is closed, \( p \in B \).

Thus, a set \( B \) is closed if and only if \( d(p, B) = 0 \) implies \( p \in B \).

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### 2. Prove: \( A \cup A' \) is closed for any set \( A \).

- \( A' \) is the set of limit points of \( A \), i.e., points where every neighborhood contains at least one point from \( A \) different from itself.

- \( A \cup A' \) contains both \( A \) and all of its limit points. Since closed sets contain all their limit points, we need to check that this union contains all limit points of itself.

- A limit point of \( A \cup A' \) must either be a limit point of \( A \) (which is already in \( A' \)) or a limit point of \( A' \), but limit points of \( A' \) are also in \( A' \) (since \( A' \) is closed). Therefore, \( A \cup A' \) is closed because it contains all of its limit points.

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### 3. Prove: \( A \cup A' \) is the smallest closed set containing \( A \).

- We already showed that \( A \cup A' \) is closed in the previous step.

- Let \( F \) be any closed set such that \( A \subseteq F \). We claim that \( F \subseteq A \cup A' \).

- If \( p \in F \), either \( p \in A \), or \( p \) is a limit point of \( A \). Since \( F \) is closed, it must contain all the limit points of \( A \), which means \( A' \subseteq F \).

- Therefore, \( F \subseteq A \cup A' \), and since \( A \subseteq A \cup A' \subseteq F \), we have \( F = A \cup A' \).

Thus, \( A \cup A' \) is the smallest closed set containing \( A \).

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Let me know if you'd like more details on any of these proofs! Here are some follow-up questions to explore:

1. Can you give an example of a closed set in \( \mathbb{R}^n \) and compute the distance function?
2. What are some properties of open sets compared to closed sets?
3. How does the closure of a set \( A \) relate to its limit points?
4. Can a set be both open and closed? If so, provide an example.
5. What is the relationship between compact sets and closed sets?

**Tip**: The closure of a set \( A \) is the smallest closed set containing \( A \), and it's always \( A \cup A' \). Understanding this helps in topology and real analysis.

Demonstrate the following:
1. A subset B of R^n is closed if and only if d(p, B) = 0 implies p ∈ B, where d(p, B) = inf{d(p, q) : q ∈ B}.
2. A ∪ A' is closed for any set A, where A' denotes the set of limit points of A.
3. A ∪ A' is the smallest closed set containing A, i.e., if F is closed and A ⊆ F ⊆ A ∪ A', then F = A ∪ A'.

This guide provides solutions to problems on proving the closure of sets and limit points in real analysis. It covers the relationship between distance functions, closed sets, and the closure of a set in R^n, including key theorems and mathematical concepts for university-level students.

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