I have read individual proofs of the interior and closure of a convex set but hoping to use one to prove the other. I want to use points on the boundary of $K$ to say that every line segment with endpoints $a,b\in int(K)$ is already contained on a line segment with end points $c,d\in bd(K)$.
Am I trying to simplify too much or can this be done? Also trying to use the proper notation since I struggle to turn sentences into the correct mathematical notation.