Equivalence of definitions of continuity One of the definitions of continuity is that the Upper Contour Set and the Lower Contour Set are closed.

I am trying to show that if preference is continuous and $x&gt;y&gt;z$, then there is some $\alpha \in [0,1]$ such that $\alpha x+(1-\alpha)z&gt;y$

Can you help me to prove second one by using closeness?
HINT: it is easier if you go through the intermediate step that the set $\{\alpha \in[0,1]: \alpha x+(1-\alpha) z \succsim y\} $ is closed for every $x,y,z$