Theorem
Let $ A $ be a set in S. Then
A ∪ S = S.
Proof
Let x ∈ A ∪ S. Then x ∈ A or x ∈ S.
If x ∈ A, then x ∈ S since A ⊆ S.
If x ∈ S, then, well, x ∈ S.
So we have that if x ∈ A ∪ S, then x ∈ S ⇒ A ∪ S ⊆ S.
Next, we want to show that S ⊆ A ∪ S.
We know this is true because the set resulting from the union of two sets contains both of the sets forming the union (proof).
Since A ∪ S ⊂ S and S ⊂ A ∪ S, we have that A ∪ S = S.
$ \blacksquare $
