Cycle Graph is 2-Regular

Theorem
Let $G$ be a cycle graph.

Then $G$ is regular.

Proof
Let $G$ be a cycle graph.

By definition, a cycle graph is a graph which consists of a single cycle $C$.

By definition, a cycle is a circuit in which no vertex except the first (which is also the last) appears more than once.

By definition, a circuit is a closed trail with at least one edge.

By definition, a trail is a walk in which all edges are distinct.

Let $v$ be a vertex of $G$.

Then $v$ is a vertex of a closed walk.

Hence for every edge which is incident with $v$, there exists another edge which is also incident with $v$.

Suppose $v$ were odd.

Then at least one edge must appear at least twice in $C$.

But then $C$ would not be a trail.

Thus $v$ is even.

Suppose $v$ has degree zero.

Then there are no edges incident with $v$.

Hence $v$ is not on a closed walk.

Hence the degree of $v$ is greater than $0$.

Suppose $v$ has degree greater than $2$.

Then $v$ must be on a circuit in which $v$ appears more than once.

Hence that circuit is not a cycle

Hence $G$ is not a cycle graph.

Hence $v$ is of degree $2$.

The result follows.