The setting is the following. I have a complex algebraic variety $X$, and $\mathcal{F}$ is a constructible sheaf on it (i.e. there is a stratification of Zariski-locally closed subsets $X=\sqcup_{i \in I} X_i$, and $\mathcal{F}$ is locally constant on each $X_i$ with respect to the euclidean topology).
My question is the following. How to show $H^i(K,\mathcal{F}_{|K})=\varinjlim_{U\supset K}H^i(U,\mathcal{F}_{|U})$? This is used in the proof of Artin-Grothendieck theorem in the book Positivity in Algebraic Geometry, by Lazarsfeld.
Both complete answers and references are welcome!