Let $G$ be a $k$-connected graph. Meaning, $G$ has no less than $k$ vertices, and for every set of $k-1$ or less vertices, if we remove them from $G$, the graph stays connected (Of course, $G$ itself is also connected).
I want to prove that for any $k>1$, if $G$ is $k$-connected, then every set of $k$ vertices is contained in a cycle.
I have tried some ways - mainly using induction by removing one of the vertices of the set from the graph, and/or using Menger's theorem to construct the cycle. But I always encounter problems with making sure that the cycle I'm building deosn't have repeating edges etc.
Help would be greatly appreciated :) Thanks!