Closed immersions preserve residue fields

By [F1], every open neighbourhood $W$ of $z$ in $Z$ is $i^{-1}U$ for an open neighbourhood $U$ of $x$ in $X$: the image $i(W)$ is open in the subspace $i(Z)$, so it is $i(Z)\cap U$ for some open $U\subseteq X$, which necessarily contains $x$. Thus [F3] identifies $(i_{\ast }{\mathcal{O}}_Z{)}_x$ with ${\mathcal{O}}_{Z,z}$. Indeed every section germ at $z$ is represented on such an $i^{-1}U$, and equality of two germs is detected on another such neighbourhood, so the identification is both surjective and injective and respects ring operations. By [F4], the surjective sheaf morphism of [F1] induces a surjection ${\mathcal{O}}_{X,x}\to (i_{\ast }{\mathcal{O}}_Z{)}_x$. Under the germ identification this is the stalk homomorphism of $i$, proving its surjectivity.

By [F2] and [F5], this stalk homomorphism induces a unital field homomorphism $\kappa (x)\to \kappa (z)$. It is surjective: represent an element of $\kappa (z)$ by an element of ${\mathcal{O}}_{Z,z}$, lift it using step 1.1 and take the residue class of that lift. It is injective: if a nonzero source element mapped to zero, its inverse would force $1=0$ in the target field, a contradiction. Hence the induced homomorphism is an isomorphism. If $Z$ is empty, the pointwise assertions are vacuous.