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**MissMousey** Problem: Prove that the union of any collection of open sets is open. Is the following proof acceptable?

Proof: Let U' denote the collection of open sets and let U' = $\displaystyle \bigcup U_n$ , where n = 1,...,k. Since U' is a collection of open sets, there exists an interior point $\displaystyle s_0$ that belongs to U'. Since $\displaystyle s_0$ belongs to U', it is implied that $\displaystyle s_0$ belongs to at least one $\displaystyle U_n$. So for $\displaystyle s_0$ that belongs to $\displaystyle U_1 \in U' $, there exists a neighborhood in $\displaystyle U_1$; For $\displaystyle s_0$ that belongs to $\displaystyle U_2 \in U' $, there exists a neighborhood in $\displaystyle U_2$,...If we take the union of all $\displaystyle U_n$ , there exists a neighborhood in U' for all $\displaystyle s_0$. Therefore, U' is open.