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Show that any closed polygonal path can be decomposed into a finite union of simple closed polygonal paths and line segments traversed twice in opposite directions.

MY SOLUTION

Suppose $\gamma(t):a\leq t\leq b$ has $\gamma (t_{2}) = \gamma (t_{1})$. Then $\gamma$ can be written as a union of $\gamma_{1}$ and $\gamma_{2}$ where $\gamma_{1}=\gamma(t); t\in[a,t_{1}]\bigcup[t_{2},b]$ and $\gamma_{2}=\gamma(t)$; $t\in[t_{1}, t_{2}]$.

Is correct my method?

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    I don't see what you are trying to claim in your approach. Read the question again, carefully. Try to draw a few increasingly more complicated closed polygonal paths, in particular non-simple ones and see if you can come up with an idea.2012-05-02

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