Eulerian Circuit is Eulerian Trail

Theorem

Let $G$ be a graph.

Let $C$ be an Eulerian circuit for $G$.

Then $C$ is also an Eulerian trail for $G$.

Proof

Recall the definition of Eulerian circuit:

An Eulerian circuit is a circuit that passes through every vertex of a graph and uses every edge exactly once.

Recall the definition of Eulerian trail:

An Eulerian trail is a trail $T$ that passes through every vertex of a graph $G$ and uses every edge of $G$ exactly once.

Recall the definition of circuit:

A circuit is a closed trail with at least one edge.

Hence an Eulerian circuit is an instance of an Eulerian trail.

$\blacksquare$