Let $G$ be a non-bipartite simple graph with its minimum degree $\delta > [2V/(2k+1)]$, where $V$ denotes the number of vertices of $G$, and $k>=2$ is an integer. Show that $G$ has odd cycles of length $<=2k-1$.
I do not know how to deal with this problem, but I have thought about the simplest situation when $k=2$. Then the extreme situation is something like nested pentagon-cycles, where each vertex of the graph has edges exactly with vertices from different vertex of pentagon. Then the graph is non-bipartite and satisfies all the conditions. And in this graph it is clear that $G$ has triangles. Please help with the general situation. thank you!