SOLUTION: Prove the triangle inequality by showing a step by step process.

The triangle inequality is the statement, saying that 


  +-------------------------------------------------------+
  |    if "a", "b" and "c" are the sides of a triangle    |
  |   on a plane, then a + b > c, a + c > b, b + c > a.   |
  +-------------------------------------------------------+


Let ABC be the triangle with vertex A opposite to side a,

vertex B opposite to side b, and vertex C opposite to side c.


Let's prove first inequality  a + b > c.

Use axiom of Geometry, saying that the interval of the straight line between points A and B 
is the shortest distance between points A and B on the plane.

It means literally that a + b > c.


Thus the triangle inequality is one step from this axiom.


The equality sign a + b = c is possible if and only if the triangle ABC is degenerated,
so point/vertex C lies on the side "a".


The proofs for other triangle inequalities a + c > b  and  b + c > a  is the same.