I am having problem with the construction in Serre's Vanishing Theorem. The proof begins with a general construction which I don't follow.
Let $X$ be a scheme. $x \in X$ a closed point, $U\subseteq X$ an affine neighborhood. $Z=X \setminus U, Z'=X \cup \{x \}$. There are quasi-coherent sheaves of ideals $I$, $I'$ cutting out $Z$ and $Z'$. Giving SES, $$ 0 \rightarrow I' \rightarrow I \rightarrow I/I' \rightarrow 0 $$
How does one obtain injectivitiy at $I' \rightarrow I$?
By definition, $I\subseteq\mathcal{O}_X$ is the subsheaf consisting of sections which vanish in the residue field at every point of $Z$, and similarly for $I'$. Since $Z'\supseteq Z$, this means $I'\subseteq I$ (a section which vanishes on $Z'$ also vanishes on the smaller set $Z$). The morphism $I'\to I$ in the exact sequence is then just the inclusion map, which is obviously injective.