Whilst studying eulerian graphs I came across the problem:
Problem Prove that a graph $G$ with at least one edge and all vertices of even degree must contain a cycle.
My attempt: If $G$ is connected, then by definition it would be an Eulerian graph (Eulerian graphs are connected with even degree), and so it would have a cycle. Now suppose that $G$ is not connected, then choose some connected component $H_1 \subset G$ where that the edge set has cardinality greater than or equal to $1$ , which exists since $G$ has at least one edge. Then this graph is connected with vertices of even degree, and so is eulerian, and hence it has a cycle.
Is this reasoning correct?
Edit: My reasoning is NOT correct since what I showed is that there is an Euler tour which is not a cycle but a closed trail for anyone reading this in the future..