The statement you've provided, \( \sup(A \cup B) = \max(\sup A, \sup B) \), is valid under certain conditions. Here's an explanation:

### Definitions:
- **Supremum (sup)**: The least upper bound of a set. If \( A \) is a subset of the real numbers, \( \sup A \) is the smallest number that is greater than or equal to every element of \( A \).
- **Union (\( A \cup B \))**: The set containing all elements that are in \( A \), \( B \), or both.

### Statement Breakdown:
The statement claims that the supremum of the union of two sets \( A \) and \( B \) is equal to the maximum of the suprema of the individual sets. In other words:
\[ \sup(A \cup B) = \max(\sup A, \sup B) \]

### Why is this true?
1. **Upper Bound**: 
   - Any upper bound of \( A \) is also an upper bound of \( A \cup B \), and the same goes for \( B \).
   - The supremum of \( A \cup B \) must therefore be at least as large as \( \sup A \) and \( \sup B \).
   - Thus, \( \sup(A \cup B) \geq \max(\sup A, \sup B) \).

2. **Least Upper Bound**:
   - The smallest number that bounds both \( A \) and \( B \) from above will be the larger of \( \sup A \) and \( \sup B \), i.e., \( \max(\sup A, \sup B) \), since it is the least upper bound that bounds both sets.

Hence, the supremum of the union is equal to the maximum of the suprema of the individual sets.

This reasoning holds if \( A \) and \( B \) are non-empty and bounded from above.

Would you like a more detailed breakdown of any part of this reasoning?

### Follow-up questions:
1. What happens if one of the sets is unbounded?
2. Can the supremum and maximum of a set ever be different?
3. How would this apply if \( A \) or \( B \) were empty?
4. Is this property of suprema true for infinite sets?
5. What is the difference between the supremum and the infimum?

### Tip:
When working with suprema, always ensure the sets in question are bounded above; otherwise, the supremum may not exist within the set of real numbers.

This problem explores the relationship between the supremum of the union of two sets and the maximum of the individual suprema. Key concepts include set theory, real analysis, and the supremum theorem. Suitable for undergraduate students learning about suprema and bounds.

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