# Generalization of Myers theorem

I need a hint to this problem, or anything that gives a pointer in the right direction would be helpful. The question I need to prove is:

Let $$M$$ be a compact $$n$$-manifold such that Ric$$(U,U) \geq (n-1)k$$ for all $$U \in TM$$ such that $$||U||=1$$. Prove that $$\gamma$$ is a geodesic in $$M$$ of length greater than $$\frac{m\pi}{\sqrt{k}}$$, then $$\gamma$$ has index at least $$m$$.

Now I have seen a proof of a similar result where if $$\gamma$$ is of length greater than $$\frac{\pi}{\sqrt{k}}$$, then $$\gamma$$ has a conjugate point. This is lemma 13.142 of Manifold and Differential Geometry by Jeffery M.Lee. However I can't see a way of using this to prove what I need. Any help is appreciated.