These three inequalities often tend to appear as a package in many textbooks about real analysis, signal processing or linear algebra. It is good to know the main reason that we see this package all the time, and to separate the role of each of these three fundamental inequalities. The overall reason is that, these inequalities are the key to generalize the facts about vectors in 2D/3D spaces and the Euclidean norm to higher dimensional spaces and (non-Euclidean) $p$-norms. Each of…