I am trying to prove $K^{**} = cl(K)$, where $K^{**}$ is the double dual of K, cl(K) is the closure of K, and K is a convex cone.
I was able to show that $cl(K)\subseteq K^{**}$ and am trying to show the other direction: $K^{**} \subseteq cl(K)$. I am thinking of proving by contradiction and argue that suppose there exists y $\in K^{**}$ such that $y \notin cl(K)$, then .... From posts online that give hints, I think I should use the separation theorem, and let the two non-empty disjoint convex sets be $\{y\}$ and cl(K). However, I am not sure how to continue from here and was hoping for some hints.
Thanks a lot.