Each path can be reduced to a simple path

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

Fugures: 

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 Figure 1: path reduction

 

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 Figure 2: two counter examples

 

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 Figure 3: The process of the proof

 

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Figure 4: Cantor-like functio

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