Let $k\geq 2$. Prove that if $G$ is $k$-regular, then $G$ contains a cycle with at least $k+1$ edges.
The way I did it was to prove that the longest path in $G$ must have at least $k$ edges, and that such a path must be one short of a cycle, and thus the longest cycle must contain at least $k+1$ edges. Is this way correct? Are there other more direct ways of approaching this problem? Thanks!