I am trying to solve the exercise 2.4 chapter III in Hartshorne's "Algebraic Geometry". For this I would like to prove for a sheaf $F$ of Abelian groups on a topological space $X$ and $U$ open subset of $X$ with complement $Z$ the following $$ H^{i}_{Z}(X,j_{.}(F\mid_{U})) = 0$$ where $j:U \longrightarrow X$ the inclusion and $j_{.}$ meaning the extension of a sheaf on U to a sheaf on X by zero. I have proven exercise 2.3 chapter III, so I can use this.

Does anyone know a nice proof for this? Or maybe a counterexample if this doesn't hold? I was able to reduce/adapt this case to proving $H^{i}(X,j_{.}(F\mid_{U})) \cong H^{i}(U,F\mid_{U})$ but I also haven't a rigorous proof for this. Also for it is quite clear because $H^{0}_{Z}(X,j_{.}(F\mid_{U})) =0$, it is just for the higher order groups I have't an argument. (Or maybe this all doesn't hold?).

Thanks in advance! (Not sure if this is supposed to be on mathstack, but there I have not yet received an answer on this).