I have a question about convex set. how can I prove that a set is convex if and only if its intersection with any line is convex. I have worked out proof but it is not that rigorous and I was wondering if someone could help me with this
if S is convex
exist x,y that belongs to S -> lambda(x) + (1-lambda) (y) belongs to S
therefore x and y are on the line L
->lambda(x) + (1-lambda) (y) belongs to L
->lambda(x) + (1-lambda) (y) belongs to intersection of L and S
therefore S interesection L is convex
there exists x,y that belongs to S intersection with L -> lambda(x) + (1-lambda) (y) belongs to S intersection with L
-> lambda(x) + (1-lambda) (y) belong to S
->S is convex