IF B is bounded above then so is a and supremam of A less than equal to suprema of B

The statement you’ve written seems to be a proposition involving two sets, \( A \) and \( B \), where \( A \subseteq B \) and we are discussing the boundedness and supremum (least upper bound) of these sets.

### Rephrased statement:

- If \( B \) is bounded above, then \( A \) is also bounded above.
- Moreover, the supremum of \( A \) is less than or equal to the supremum of \( B \).

### Proof Outline:

1. **Boundedness of \( A \)**:  
   If \( B \) is bounded above, it means there exists some \( M \in \mathbb{R} \) such that \( b \leq M \) for all \( b \in B \). Since \( A \subseteq B \), every element of \( A \) is also an element of \( B \). Thus, \( a \leq M \) for all \( a \in A \). Therefore, \( A \) is also bounded above by \( M \).

2. **Relationship of Supremums**:  
   Since \( A \subseteq B \), the least upper bound (supremum) of \( A \), denoted \( \sup A \), cannot be greater than the least upper bound of \( B \), denoted \( \sup B \). Formally:
   \[
   \sup A \leq \sup B
   \]
   This holds because any upper bound of \( B \) is also an upper bound of \( A \) (since \( A \subseteq B \)), and \( \sup B \) is the least such upper bound.

### Conclusion:

- \( B \) being bounded above implies \( A \) is also bounded above.
- The supremum of \( A \) is less than or equal to the supremum of \( B \), i.e., \( \sup A \leq \sup B \).

Would you like further details or any clarifications on this? 

### Related Questions:

1. What is the difference between supremum and maximum?
2. Can the supremum of a set be an element of the set?
3. What happens if \( A \) and \( B \) are equal? What can be said about their supremums?
4. How do we formally define a bounded set in real analysis?
5. What is the significance of the least upper bound property in the real numbers?

### Tip:

Always remember that the supremum of a set may not be in the set itself, but it is always the smallest possible upper bound.

IF B is bounded above then so is A and supremum of A is less than or equal to supremum of B.

Explore the relationship between bounded sets and their supremums. If set B is bounded above and A is a subset of B, this guide explains how the supremum of A is less than or equal to the supremum of B. Suitable for students in undergraduate real analysis.

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