Let $X$ be a topological vector space and $K$ a convex subset of $X$.
Then if $x \in int(K)$ ,$y\in K$ the line segment $[x,y) $ is contained in $int(K)$.
Ok my attempt is to assume that there exists $z\in [x,y)$ that is not in $int(K)$ and since this set is open and convex there should exist a continous linear functional $f$ ,and a real number $C$ such that $$sup_{int(K)}f<C\leq f(z)$$ and then approach $f(z)$ with a sequence in a way as to disrupt the continuity of $f$. I feel something is wrong in my attempt...any hint would be great.