Question

Prove that if A and B are each bounded above in R, then so is
their union.

Answer 1

To prove that the union of two bounded above sets, A and B, is also bounded above, we need to show that there exists a number that is greater than or equal to every element in the union of A and B. Let MA be an upper bound for set A and MB be an upper bound for set B. By considering the elements in the union A ∪ B, we can conclude that max(MA, MB) is an upper bound for the union of A and B.

Step-by-step explanation:

In order to prove that the union of two bounded above sets, A and B, is also bounded above, we need to show that there exists a number that is greater than or equal to every element in the union of A and B.

1. Let MA be an upper bound for set A, which means that every element in A is less than or equal to MA.
2. Similarly, let MB be an upper bound for set B.
3. Now, consider the union of A and B, denoted by A ∪ B. For any element x in A ∪ B, x can either be an element of A or an element of B.
4. If x is an element of A, then x ≤ MA, since MA is an upper bound for A.
5. If x is an element of B, then x ≤ MB, since MB is an upper bound for B.
6. Therefore, for every element x in the union A ∪ B, we have x ≤ max(MA, MB). So, the number max(MA, MB) is an upper bound for the union of A and B.