In this paper, the author focuses on the teaching-level and learning-level of the completeness axiom and its applications on [0,1] and $mathbb{R}$, $mathbb{R}{ imes}mathbb{R}$, $mathbb{R}{ imes}mathbb{R}{ imes}mathbb{R}$ by (expected) teachers in the school mathematics, which is usually introduced in the class of real analysis of university mathematics. Firstly the author considers the properties of the completeness axiom and its 19 equivalent theorems, next he deals with its importances in the school mathematics and finally he suggests the teaching and learning of the concepts on the completeness axiom and its applications on [0,1] and $mathbb{R}$, $mathbb{R}{ imes}mathbb{R}$, $mathbb{R}{ imes}mathbb{R}{ imes}mathbb{R}$ by (expected) teachers in the school mathematics.