Each path can be reduced to a simple path

Recently, I made a small conclusion, but I found it is found and well-founded in some textbook. So I post my paper here. 

• Each_path_can_be_reduced_to_a_simple_path(20181124).pdf

Fugures: 

 Figure 1: path reduction

 

 Figure 2: two counter examples

 

 Figure 3: The process of the proof

 

Figure 4: Cantor-like functio