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proof. Let $A$ and $B$ both be convex sets and $A \cap B \ne \emptyset$. Then suppose that $p,q \in A\cap B$. Since $p,q \in A \cap B$, it follows that $p,q \in A$, a convex set. Then $p + (1-\lambda)q \in A$. Since $p,q \in B$, also a convex set by assumption, $p+(1-\lambda)q \in B$. Thus, $p + (1-\lambda)q \in A \cap B$. QED

I wanted to know if everything looks okay, or if anything needs to be improved?

Skm
  • 2,392
  • Also https://math.stackexchange.com/questions/26316/what-is-the-right-way-to-prove-that-the-intersection-of-an-infinite-number-of-co?rq=1, https://math.stackexchange.com/questions/2125944/prove-that-if-s-and-t-are-convex-sets-s-cap-t-is-a-convex-set?rq=1 –  Dec 07 '18 at 22:24
  • For me it's fine – Viera Čerňanová Dec 07 '18 at 22:26
  • This proof is correct, there is nothing to change. – dallonsi Dec 07 '18 at 22:35

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