# Closed-valued and Upper Hemicontinuous

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 $$$$\phi (x) = \phi _1(x)\cap \phi _2(x) \quad \forall x\in X.$$$$ Show that $$\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?