A set in is called convex if for every pair of points and in and every real ,we have
1. Prove that the interior of convex sets is convex.
My attempt: Let be the set of interior points of . Choose , this means that such that . Now choose two points in this open ball, and since they are also interior points of . And since every open ball is convex, and thus , as required.
I have a problem with that "proof" since I never used the convexity of S, I basically proved that any interior set is convex. Or that every open set is convex. But I can't find where I'm going wrong, or a different proof that makes the use of the convexity of S.
2. Prove that the closure of a convex set is convex.
My attempt, What I have so far: Let be the closure of . Pick arbitrary , I need to show .
(i) ; I showed the implication from here very easily.
(ii) , where is the set of all the limit points of .
(iii) Without loss of generality, and .
I'm having problem with the last "2" cases as I don't know how to proceed from the fact that something is a limit point to showing that .
Any suggestions would be appreciated. Thanks, in advance.