A scheme $X$ is normal, if stalks $\mathcal{O}_{\mathfrak{p}}$ are integral closed for all $\mathfrak{p}\in X$.
A ring $A$ is normal, if it's integrally closed.
I want to show if $X$ is a normal, integral scheme, then for all open subset $U\subseteq X$, the section $\Gamma(U,\mathcal{O}_X)$ is also normal. In particular, the global section $\Gamma(X,\mathcal{O}_X)$ is normal, I really need this conclusion.
Thanks for any help!