Let $X$ be a topological space and let $U_n \subset U_{n+1}$ be an increasing sequence of open subsets of $X$. Let us note $U = \cup_n U_n.$
Do we have $$\varinjlim_{n \to + \infty} \mathbb{C}_{U_n} \simeq \mathbb{C}_U ?$$
Here is my try :
There is a well defined map $\mathbb{C}_{U_n} \to \mathbb{C}_{U}$ for all $n$, hence there is a morphism of sheaves $$\varinjlim_{n \to + \infty} \mathbb{C}_{U_n} \to \mathbb{C}_U.$$ Hence I can prove the iso on fibers. I thus have to prove that
$$\varinjlim_{n \to +\infty} \begin{cases} \mathbb{C}\,\,\, if \,\, x \in U_n \\ 0 \,\, else \end{cases} \simeq \begin{cases} \mathbb{C}\,\,\, if \,\, x \in U \\ 0 \,\, else \end{cases}.$$ This seems clear to me but I have a doubt.