I am a bit stuck with the following assertion:
Let $X$ be a separated integral scheme. Then to every (schematic) point $x \in X$ we can correspond its local ring, and look at it as a subring of the field of rational functions on $X$. How do we show that different points give rise to different such subrings?
If $X$ is affine, it is clear, since we can find a function which is zero on one point and non-zero on another. If $X$ is not affine, I guess that for two points which do not lie in same affine open subscheme we should choose affine neighbourhoods and consider their intersection, which is affine by separatedness. But what to do then?