for the following proof :
Write the theorems involved in the proof and how are they involved
(Read the note in the end.)
This seems like a nice exercise. Obviously, the reasoning displayed uses several things: (1) properties of addition, subtraction, multiplication and division; (2) properties of inequalities; (3) properties of the function; and (4) possibly some other properties, like those of square root, etc.
Imagine that you are talking to a elementary-school student who is extremely smart and understands everything you say provided you explain every new concept. However, he/she knows only the properties of the four basic operations and inequalities, as well as the definition of the absolute value. As I understand, first you are supposed to write this reasoning in much finer detail, on the level something like: "Now add ... to both sides of inequality. Now move this term to the other side. Now use the formula for the expansion of the square of a sum". Then you have to note every place that the young student won't understand without more explanation. For example, you replace with because you claim they are equal. This is not a property of the basic operations or inequalities; it involves some reasoning (very simple) about . So you have to identify all such places and write the statements that you use, such as .
Note: this is my interpretation of the problem. You may have recently studied concrete properties or theorems, to which you gave names, etc. Then maybe the professor wants you to identify those facts only.
What it does is to prove the triangle inequality.
We start very basic properties of the distance function (the metric, the absolute value) and then using basic properties of order we prove the triangle inequality.
This basic form of the proof is used in a variety of different courses.