Jtdylktuy

Let $M$ be a Riemannian manifold, $p in M$ and $C_m(p)$ shall denote the cut locus of $p$.
In his „Riemannian Geometry“ Do Carmo says that $exp_p$ is injective on a open ball $B_r(p)$ of radius $r$ if and only if $rleq d(p,C_m(p))$. I have however trouble proving the if direction.

He says that this is a consequence of the fact that if $q notin C_m(p)$, then there is unique minimizing geodesic from $p$ to $q$. He also proved the fact that if $gamma(t_0)$ is the cut point of $p=gamma(0)$ along $gamma$, then either $gamma(t_0)$ is the first conjugate point of $p$ along $gamma$ or there exists a different geodesic of equal length connecting $p$ and $gamma(t_0)$.

As I said I have trouble proving that if $exp_p$ is injective on $B_r(p)$, then $rleq d(p,C_m(p)).$
My attempt:
Assume $r>d(p,C_m(p)).$ Choose a point $q in C_m(p)$ with $d(p,q)<r$ and let $gamma$ denote a minimizing geodesic joining $p$ and $q$. If there is a different geodesic of same length joining these two points, then $exp_p$ is not inhective on $B_r(p).$ But I can‘t come up with a contradiction when $q$ is the first conjugate point of $p$ along $gamma.$

I would appreciate a hint on how to proceed or a proof.

I was able to prove
begin{equation}
d(p,C_m(p))=sup{r>0: exp_p text{is a diffeo on} B_r(p)}.
end{equation}
Does this help?