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**3. (a) Define a limit point of a subset \( A \) of a metric space \( X \). Prove that if \( x \) is a limit point of \( A \), then every neighborhood of \( x \) contains infinitely many points of \( A \).**

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Here are 5 related questions to expand on this concept:

1. What are the main differences between limit points and isolated points?
2. How can limit points be characterized in a topological space rather than a metric space?
3. What role do limit points play in the closure of a set?
4. Can a limit point of a set in a metric space belong to the set's complement?
5. How are limit points used in the context of sequences in metric spaces?

**Tip:** A point \( x \) is a limit point of a set \( A \) if every neighborhood around \( x \) contains at least one point of \( A \) other than \( x \) itself. This is a key idea in both topology and analysis.

Define a limit point of a subset A of a metric space X. Prove that if x is a limit point of A, then every neighborhood of x contains infinitely many points of A.

Explore the definition of a limit point in a metric space and learn how to prove that every neighborhood of a limit point contains infinitely many points of the subset. This concept is fundamental in topology and real analysis, suitable for undergraduate students.

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