Let $K$ be a field, and $R$ an integrally closed subdomain of $K$ with $K$ as it's quotient field. Let $(R_{\alpha})$ be a family of valuation rings of $K$ with $\cap_{\alpha}R_{\alpha}=R$. Show that the integral closure of $R$ in $L$, an extension field of $K$, is the intersection of all valuation rings of $L$ which dominate one of the $R_{\alpha}$.
I have been trying it for so long now but unable to solve the problem. So any help will be appreciated about this problem. Thanks