Any edge that occurs an odd number of times in a closed walk $W$ is part of a cycle in $W$ I saw this result in this question:
Any edge that occurs an odd number of times in a closed walk $W$ is part of a cycle in $W$.
From the questions in my introductory graph theory class, this seems like a useful result, and I wanted to prove it as an exercise. It was more difficult than I thought, I'm struggling with defining/using what it means for an edge to occur an odd number of times.
Is this result even true, and if it is, how can it be proved?
 A: Let the walk be $v_1,v_2\ldots,v_t$, where $v_i,v_{i+1}$ are adjacent vertices and $v_t=v_1$. For an edge $uv$ to appear $k$ times means that there exist $k$ distinct indices $j$ such that: $\{v_j,v_{j+1}\}=\{u,v\}$. This edge can appear in the order $uv$ or $vu$. 
Do the following: while there exist at least two occurrences of the edge $uv$, do: If the order of some two consecutive appearances of this edge is same: $u,v,P,u,v$ (where $P$ is the part of the walk between consecutive appearances), then replace this segment of the walk ($uvPuv$) with just $uv$; this ensures that the resulting sequence is still a closed walk. Similarly, if the order is $u,v,P,v,u$, replace this with just the vertex $u$. When this process can no longer be repeated, we get a closed walk with the number of occurrences of $uv$ equal to 1.
Now in this reduced closed walk, consider a closed sub-walk $W'$ with smallest number of edges and containing $uv$. We can see that no edge can appear twice in $W'$; If some edge say $xy$ appears at least twice, we can eliminate it. If $xyPxy$ is a subwalk, then consider cases where $P$ contains $uv$ and $P$ does not; in both cases, we can find a shorter closed walk containing $uv$. Similarly if $xyPyx$ is a subwalk.
Thus we get a closed walk containing $uv$ with edges from $W$, and every edge present exactly once, that is a cycle containing $uv$.
