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The image contains four mathematical statements that require proof. I will explain and provide the proofs one by one:

1. **Demonstrar: Si \( A \) es un subconjunto finito de \( \mathbb{R} \), entonces el conjunto derivado \( A' \) de \( A \) es vacío, i.e. \( A' = \emptyset \).**

   **Proof:**
   A subset \( A \subset \mathbb{R} \) is finite if it contains a finite number of elements, say \( A = \{a_1, a_2, \dots, a_n\} \). The derived set \( A' \) of a set \( A \) is the set of all limit points of \( A \). 

   In \( \mathbb{R} \), a limit point of a set is a point \( x \) such that every neighborhood of \( x \) contains at least one point of \( A \) different from \( x \) itself. However, since \( A \) is finite, there is no such point, as neighborhoods of any element of \( A \) will not contain any points of \( A \) other than the point itself. Thus, no point in a finite set can be a limit point.

   Therefore, \( A' = \emptyset \).

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2. **Demonstrar: Todo subconjunto finito de \( \mathbb{R} \) es cerrado.**

   **Proof:**
   A set \( A \subset \mathbb{R} \) is closed if it contains all its limit points. As demonstrated in the previous proof, a finite subset of \( \mathbb{R} \) has no limit points, i.e., \( A' = \emptyset \). Since there are no limit points to include, the set trivially contains all its limit points.

   Therefore, every finite subset of \( \mathbb{R} \) is closed.

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3. **Demonstrar: Si \( A \subset B \), entonces \( A' \subset B' \).**

   **Proof:**
   Let \( A \subset B \), meaning that every point in \( A \) is also a point in \( B \). The derived set \( A' \) consists of the limit points of \( A \), and \( B' \) consists of the limit points of \( B \).

   Now, consider a point \( x \in A' \), meaning that every neighborhood of \( x \) contains a point of \( A \) different from \( x \). Since \( A \subset B \), every point of \( A \) is also a point of \( B \). Therefore, every neighborhood of \( x \) also contains a point of \( B \) different from \( x \), implying that \( x \) is a limit point of \( B \).

   Hence, \( x \in B' \), which means \( A' \subset B' \).

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4. **Demonstrar: Un subconjunto \( B \) de \( \mathbb{R} \) es cerrado si y solo si \( d(p, B) = 0 \) implica \( p \in B \), donde \( d(p, B) = \inf\{d(p, q) : q \in B\} \).**

   **Proof:**
   - (⇒) Suppose \( B \subset \mathbb{R} \) is closed. This means \( B \) contains all its limit points. Consider a point \( p \in \mathbb{R} \) such that \( d(p, B) = 0 \), meaning that for any \( \epsilon > 0 \), there exists a point \( q \in B \) such that \( d(p, q) < \epsilon \). In other words, every neighborhood of \( p \) contains a point of \( B \). This implies that \( p \) is a limit point of \( B \). Since \( B \) is closed, it contains all its limit points, so \( p \in B \).

   - (⇐) Now, suppose \( d(p, B) = 0 \) implies \( p \in B \). Consider any limit point \( p \) of \( B \). By definition, every neighborhood of \( p \) contains a point of \( B \), meaning \( d(p, B) = 0 \). By assumption, this implies \( p \in B \), showing that \( B \) contains all its limit points.

   Therefore, \( B \) is closed.

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If you would like further clarification or a more detailed explanation of any part, feel free to ask! Here are five follow-up questions to expand on these concepts:

1. What is the definition of a derived set in general topological spaces?
2. Can a countably infinite set in \( \mathbb{R} \) have a non-empty derived set?
3. How do closed sets relate to compact sets in \( \mathbb{R} \)?
4. Is the intersection of closed sets always closed? Why or why not?
5. What happens to the closure of a set when it is mapped under a continuous function?

**Tip:** In \( \mathbb{R} \), closed sets are closely related to compactness, but a set can be closed without being compact (e.g., \( [0, \infty) \)).

Demonstrate the following:
1. If A is a finite subset of R, then the derived set A' of A is empty, i.e., A' = ∅.
2. Every finite subset of R is closed.
3. If A ⊆ B, then A' ⊆ B'.
4. A subset B of R is closed if and only if d(p, B) = 0 implies p ∈ B, where d(p, B) = inf{d(p, q) : q ∈ B}.

Explore key concepts in real analysis, including proofs that finite subsets of R are closed, derived set relationships, and characterizations of closed sets using distance functions. Suitable for university-level students.

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