a monomorphism of schemes induce homomorphism $\kappa ( f ( x )) → \kappa ( x )$ is an isomorphism. Show that a monomorphism $f : X \rightarrow S$ of schemes is separated, purely
inseparable(i.e. universally injective) and for $x \in X$ the induced homomorphism $\kappa ( f ( x )) \rightarrow \kappa ( x )$ is an isomorphism.
This is exercise 9.6 in Görtz & Wedhorn's book.
Since $f$ is monic, the diagonal $\Delta_f$ is an isomorphism, so $f$ is separated and universally injective($\Delta_f$ is surjective).
But I don't know how to show that  $\kappa ( f ( x )) \rightarrow \kappa ( x )$ is an isomorphism. Any help would be appreciated!

Since being a monomorphism is preserved by base change, we can base change along $\operatorname{Spec} k(f(x))\to S$ and assume that $S$ is the spectrum of a field. Next, since open immersions are monomorphisms, we may replace $X$ by an affine open neighborhood of $x$ and assume $X$ is affine. Thus we've reduced to the case where $\operatorname{Spec} R\to \operatorname{Spec} k$ is a monomorphism, which implies that it's diagonal map is an isomorphism. But $R\to R\otimes_k R$ can only be an isomorphism if $\dim_k R \leq 1$, and as $R$ admits a nonzero map to $k(x)$, it is nonzero and we have the result.
