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The problem is from Berkeley Econ 204, question 6.

Let $X \subset E^n $ and $Y\subset E^m$, $\phi _1$, $\phi _2$ : $X\to 2^Y$ be closed-valued upper hemicontinuous correspondences. Suppose that $\phi _1(x)\cap \phi _2(x) \neq \varnothing$ for each $x\in X$ and define the correspondence $\phi: X\to 2^Y$ by \begin{equation} \phi (x) = \phi _1(x)\cap \phi _2(x) \quad \forall x\in X. \end{equation} Show that $\phi$ is upper hemicontinuous.

I can understand the solution given in the link, but it seems hard and tricky to construct the desired open set directly... So is there any other proofs?

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