Proving $\sup A=\sup B,\;A\subseteq\mathbb R,\; B =\{\frac{x+y}{2}| x,y\in A \}$

Question

How can the below question be proven? Is B the set of the average of every possible sum of x and y values in A?

Question

Answer 1

Let $s=\sup A$ and $t= \sup B.$

1. If $z \in B$ , then there are $x,y \in A$ such that $z=(x+y)/2.$ Hence $z \le (s+s)/2=s.$ This gives $t \le s$.

2. If $x \in A$, then $x=(x+x)/2 \in B$, thus $x \le t.$ This shows that $s \le t.$