Let 𝑆 be a non-empty bounded closed set in ℝ. Let 𝑏 = sup(𝑆). Prove 𝑏 ∈ 𝑆

To prove that \( b \in S \) for a non-empty bounded closed set \( S \subset \mathbb{R} \), where \( b = \sup(S) \) (the supremum of \( S \)), we proceed as follows:

### Proof:

1. **\( b = \sup(S) \):**  
   By definition, \( b = \sup(S) \) means that:
   - \( b \) is an upper bound of \( S \), i.e., \( x \leq b \) for all \( x \in S \).
   - For every \( \epsilon > 0 \), there exists some element \( x_\epsilon \in S \) such that \( b - \epsilon < x_\epsilon \). This property follows from the definition of the supremum as the least upper bound.

2. **Suppose \( b \notin S \) (for contradiction):**  
   Assume, for the sake of contradiction, that \( b \notin S \). Since \( S \) is closed, this would mean that \( b \) is not a limit point of \( S \) either.

3. **\( S \) is closed:**  
   Since \( S \) is closed, it contains all its limit points. Therefore, if \( b \) is a limit point of \( S \), it must belong to \( S \). However, we assumed that \( b \notin S \), so \( b \) cannot be a limit point of \( S \).

4. **Contradiction from the supremum definition:**  
   Given that \( b = \sup(S) \), for every \( \epsilon > 0 \), there exists \( x_\epsilon \in S \) such that \( b - \epsilon < x_\epsilon \leq b \). As \( \epsilon \to 0 \), the values of \( x_\epsilon \) get arbitrarily close to \( b \). This implies that \( b \) is a limit point of \( S \). Since \( S \) is closed, it must contain all its limit points, including \( b \).

5. **Conclusion:**  
   The assumption that \( b \notin S \) leads to a contradiction. Therefore, \( b \in S \).

Thus, we have proved that \( b = \sup(S) \in S \) when \( S \) is a non-empty bounded closed set in \( \mathbb{R} \).

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### Related Questions:
1. What is the definition of a supremum in general sets?
2. How does the property of closed sets ensure all limit points are included?
3. Can the result be generalized to unbounded sets in \( \mathbb{R} \)?
4. How does the concept of infimum relate to this proof?
5. What would happen if the set \( S \) was not closed?

**Tip:** Always verify if a set is closed when working with suprema or infima to ensure limit points are considered properly.

Let S be a non-empty bounded closed set in ℝ. Let b = sup(S). Prove b ∈ S

Explore the proof that if S is a non-empty bounded closed set in ℝ, and b = sup(S), then b belongs to S. This problem covers concepts from real analysis, including supremum properties, closed sets, and limit points. Ideal for undergraduate mathematics students.

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