3.  \[it:Js+t\] $J(s+t)=0$ for some $t\in [\delta,k]$.

Given a closed geodesic $\gamma\in\Lambda M$ and $s\in S^1$, let us denote by $\gamma_s : [0,1]\to M$ the geodesic path $\gamma_s(t) := \gamma(s+t)$. Since $\operatorname{\overline{ind}}(c)>0$, Bott iteration inequality (\[eq:Bott\]) and the concavity bound (\[eq:conc\]) imply that there exists $k\in{{\mathbb{N}}}^*$ such that $$\operatorname{ind}_\Omega((c^k)_s) \geq 1,
    \quad \forall s\in S^1.$$ Let us fix such a $k\geq 1$. Then according to the Morse index theorem (\[eq:MorseIndexThm\]) for all $s\in{{\mathbb{R}}}$ we can find a non-zero Jacobi field $J$ of $c$ satisfying conditions (\[it:Js\]) and (\[it:Js+t\]). Let $r>0$ be the injectivity radius along the curve $c$. We define $\delta>0$ by $$\delta = \frac{r}{|\dot{c}(0)|},$$ (we recall that $c$ has constant speed $|\dot{c}(0)|$). Then by definition of $r$, $\delta < 1$. Given any Jacobi field $J\in{{\mathbb{J}}}(c)\setminus 0$, there exists a smooth family of geodesics $(\gamma_s)$ with $\gamma_0=c$ such that $J=\partial_s\gamma_s|_{s=0}$. Suppose that $J(0)=J(t)=0$ with $t<\delta$, then for some $s$ close to $0$, $\gamma_s$ must intersect $c$ at $c(t_0)$ and $c(t_1)$ with $2|t_0|$ and $2|t_1-t|$ strictly less than $\delta-t$ so that $|t_1-t_0|<\delta$ which contradicts the definition of the injectivity radius $r$. Hence $\delta$ fulfills the condition of the lemma for any non-zero Jacobi fields vanishing at $0$.

In order to fix 