Show that the supremum of a collection of lower semicontinuous function is lower semicontinuous

real-analysis continuity semicontinuous-functions

Fix any $x \in \Bbb{R}^n$. By definition of $\sup$ you have $$\sup_{\alpha \in I} f_{\alpha} (x) > t \ \ \Longleftrightarrow \ \ \exists\alpha \in I : f_{\alpha} (x) > t$$

Hence $$x \in S_g(t) \ \ \Longleftrightarrow \ \ g(x) > t \ \ \Longleftrightarrow \ \ \exists\alpha \in I : x \in S_{f_{\alpha}} (t) \ \ \Longleftrightarrow \ \ x \in \bigcup_{\alpha \in I} S_{f_{\alpha}} (t)$$

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