My original goal was to show for affine curves over $\mathbb{C}$, being normal is the same as smooth. For this, there are two pieces of lemma: smooth->normal, and normal implies singular points has codim >=2. I could prove the latter one with elementary tools.
The definition I am working over with are
Smooth at a point if the local ring is a regular local ring (i.e. dim m/m^2=dim A).
Normal if every localization at maximal ideal is integrally closed.
I don't see how dimensions are connected to being integrally closed.