Proof that a Polytope has vertices

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As part of my Discrete Optimization course, I have a homework where I have to prove that a Polytope has vertices. I seems to have all tools in hand (definition of a vertex, polytop, convex hull, etc.) but I am struggling to find "where to start".

Here is what is asked :

homework http://imageshack.com/a/img838/514/xz4y.jpg

If anyone would have hints, ideas, that would really help !

thank you