Trying to show that the set $S = \{(x_1,x_2):x_1x_2\geq4,x_1>0,x_2>0\}$ is convex.
I tried proving by definition, let $x=(x_1,x_2)$ and $y=(y_1,y_2)$ and $\lambda\in[0,1]$ We want to show that $\lambda x + (1-\lambda)y \in S$
for $\lambda=1$ and $\lambda=0$ the answer is trivial. for $\lambda \in (0,1)$ I got a bit stuck
What I tried was
$\lambda x + (1-\lambda)y = \lambda(x_1,x_2) + (1 - \lambda)(y_1,y_2) = \lambda(x_1,x_2) - \lambda(y_1,y_2) + (y_1,y_2) = (\lambda x_1 - \lambda y_1 + y_1, \lambda x_2 - \lambda y_2 + y_2)$
So now if we can prove that $(\lambda x_1 - \lambda y_1 + y_1)\times(\lambda x_2 - \lambda y_2 + \lambda y_2) \geq 4$ we are done.
I got stuck here, I tried looking for a way with the mean inequality but with no success. I think I am way off base, perhaps Jensen's inequality or something in that domain might be what I am looking for.