Math Help - graph theory

1. graph theory

prove that a closed walk of odd length contains a cycle.

i am able to proceed upto this stage:

a closed walk may be like this v1,v2,........,v1

now,if all the points are distinct in the walk,then the given walk represents a cycle.

& if suppose a point,say v, is the first point in the walk to be repeated,then if we start from the first v to the repeated v then it will form a cycle.

but i am not able to understand why a closed walk of odd length is required.

2. Consider the walk $v_1v_2v_1$.

3. @above

but then again,

if we consider a walk v1,v2,v3,v2,v4,v1 like this,then it will again not have any cycle.

so,i think v2,v3,v2 is to be considered a cycle of length 1 or 2,even though it is not defined.