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?