I need to prove that the the set of all averages of two numbers in $A$ is compact. When $A$ itself is compact. ie {$\frac{a+b}{2}: a,b \in A$} is compact.
Whats the best way? To argue for the converging sequences a and b that converges and then $c = \frac{a + b }{2}$ is a new sequence that has a sub sequence that also converges in $A$
Or to go through the closed and bounded properties ?
Any hints?
Consider that map $a\colon A\times A\longrightarrow\mathbb R$ defined by $a(x,y)=\frac{x+y}2$. Then $a(A\times A)$ is compact, since $A\times A$ is compact (because we're assuming that $A$ is compact) and $a$ is continuous.