Any finite Graph G with all V have at least degree of 2, is it true that every vertex is necessarily contained IN a cycle?

Question

As title, (note: this questions is asking weather or not all vertices are contained IN a cycle not asking if the G contains a cycle.

My attempt is that:

So this graph would be an counter example that disprove that every vertex is contained in a cycle. Would like to know if this is a valid counter-example since V1 is not contained in a cycle.

PS: As I was finding examples, also observed that if the all V have exactly degree of 2, then it is necessary that all vertices are contained in a cycle. Please prove if I am missing something.