Question

Let s and t be nonempty bounded subsets of r.

(a) prove if s ⊆ t , then inf t ≤ inf s ≤ sup s ≤ sup t .
(b) prove sup(s ∪t ) = max{sup s,sup t }. note: in part (b), do not assume s ⊆ t .

Answer 1

In part (a), if s is a subset of t, then the infimum of t is less than or equal to the infimum of s, and the supremum of s is less than or equal to the supremum of t. In part (b), the supremum of the union of sets s and t is equal to the maximum value between the supremum of s and the supremum of t.

Step-by-step explanation:

(a) To prove that inf t ≤ inf s ≤ sup s ≤ sup t, we need to show that the infimum of set t is less than or equal to the infimum of set s, the supremum of set s is less than or equal to the supremum of set t.

Since s ⊆ t, any lower bound of t will also be a lower bound of s, meaning inf t ≤ inf s. Similarly, any upper bound of s will also be an upper bound of t, meaning sup s ≤ sup t.

(b) To prove that sup(s ∪ t) = max{sup s, sup t}, we need to show that the supremum of the union of sets s and t is equal to the maximum value between the supremum of set s and the supremum of set t.

Let's assume sup s = M and sup t = N. The union of sets s and t will contain both sets, so any upper bound of s or t is also an upper bound of the union. Therefore, the supremum of the union will be the maximum value between M and N, i.e., max{sup s, sup t}.