A simple problem about 2-factor in graph theory

Question

For a given graph G（simple,no direction,connected）,if every vertex has an even degree, then G has a 2-factor（i.e. there are edge-distinct cycles covering all vertices）. I think I've ever seen this before but I didn't find a proof. Could you please help me with this？

Answer 1

By induction of $E(G)$, the number of edges. Let $G$ be a graph as in your question. Then $G$ has a cycle (To show this consider a maximal path in $G$, it must be cycle.) say $C$. Let $F=G\setminus E(C)$. It clear that $F$ is still an even graph, however it may not be connected. Apply the induction hypothesis to each component of $F$ and you're done.