Hinge loss is the tightest convex upper bound on the 0-1 loss

Question

I have read many times that the hinge loss is the tightest convex upper bound on the 0-1 loss (e.g. here, here and here). However, I have never seen a formal proof of this statement. How can we formally define the hinge loss, 0-1 loss and the concept of tightness between functions to prove this statement?