length less than $\varepsilon$ for $0\leq i\leq j-1$.

Let $\Omega$ be a finite-dimensional approximation of $\Lambda M$ and $C\subset
M$ a locally convex set with compact boundary such that $C \subset
\mathcal{O}$. By the local convexity of $C$, there exists an $\varepsilon>0$ such that for two points $p,q \in
C$ with Riemannian distance $d(p,q)<\varepsilon$, there exists a unique geodesic of length $=d(p,q)$ joining $p$ and $q$, and contained entirely in $C$. The negative gradient of the restriction of the energy functional to $\Omega$ is given by $$-\nabla E|_{\Omega}(\gamma) = -2
    \big(\dot{\gamma}_1(1/j)-\dot{\gamma}_2(1/j),\ldots,
    \dot{\gamma}_{j-1}((j-1)/j)-\dot{\gamma}_{j}((j-1)/j)\big)$$ for $\gamma \in \Omega$, where $\gamma_i = \gamma|_{[(i-1)/j,i/j]}$ for $1 \le i \le j$ (see [@GM68 p. 252]). Now from our choice of $j$ and Cauchy-Schwarz inequality, we get $$\begin{aligned}
    d(\gamma((i-1)/j),\gamma(i/j))^2 \le
    \frac{1}{j}E(\gamma|_{((i-1)/j,i/j)}) \le
    \frac{\varepsilon^2}{\kappa}\kappa=\varepsilon^2\end{aligned}$$ and consequently by local convexity of $C$, the negative gradient flow of the finitedimensional approximation of the energy functional respects $\Omega$.

Index of a closed geodesic
--------------------------

The index of a closed geodesic $\gamma$ is the Morse index of $E$: $$\operatorname{ind}(\gamma) := \operatorname{ind}(E,\gamma).$$ It is always finite. The behavior of this index under iteration $k\mapsto \operatorname{ind}(\gamma^k)$ was extensively studied by Bott in [@Bot56]. We simply recall that $$\label{eq:Bott}
    \operatorname{ind}(\gamma^k) \geq k\operatorname{\overline{ind}}(\gamma) - \dim(M) 