Image of the map $Spec\mathcal O_{X,x}$ to $X$ Let X be scheme and $x\in X$. Can someone help to show that the image of the morphism $Spec \mathcal O_{X,x}\rightarrow X$ is the set of points of X that specialize to x. And how does it follow form the above fact that: Set of prime divisors of X containing x and set of prime divisors of $Spec \mathcal O_{X,x}$ are in one-to-one correspondence.

In the affine case, this is the well-known classification of prime ideals of a localization of a commutative ring at a prime ideal. In the general case, glue everything.