Let $X$ be noetherian topological space and let ${I_\alpha}$ be direct system of injective sheaves of abelian groups on $X$. This is question 2.6 of chapter 3 of Hartshorne. Hint given in the book is as follows- First show that a sheaf $I$ is injective if and only if for every open set $U$ $\subset$ $X$ , and for every subsheaf $R$ $\subset$ $Z_U$, and for every map $f:R \rightarrow I$ ,there exist an extension of $f$ to a map of $Z_U$ $\rightarrow$ $I$ .
I tried to prove by Zorns lemma. But I am not able to show that maximal element is the required extension. Please someone help.
Edit- This is how I approached the problem .
Let $i$ be the inclusion of sheaf $F$ in sheaf $G$ and $f$ be morphism from sheaf $F$ to $I$. Let $A$ = {$f’$:$ F’ \rightarrow I$ | $F \subseteq F’ \subseteq$ $G$ }. By Zorns lemma family $A$ has maximal element. Let the maximal element be $f’$:$F’$ $\rightarrow$ $ I$ . Can someone help me to show that $F’$ is in fact $G$.