integers $m\ge m_0$ and for all $i\in\{1,\ldots,k\}$ the following holds:

i)  \[it:existsU\]There exists a neighborhood $U$ of $S^1\cdot\gamma_i^m$ in $\Omega$ such that $U \cap E^{-1}([0,E(\gamma_i^m)))$ is connected.

Set $A:=\max\{E(\gamma_i^{m_0})\,|\, i\in\{1,\ldots,k\}\}$, and notice that $A$ is larger than the energy of a closed geodesic of mountain pass type. We fix an identification of $\pi_1(M)$ with ${{\mathbb{Z}}}$ and denote by $[\gamma]\in{{\mathbb{Z}}}$ the class of a loop $\gamma\in\Lambda M$. We define the following sets of curves: $$\begin{aligned}
        P_j^\pm := \{\gamma \in \Omega\,|\, \gamma(S^1) \subset
        \text{int}(C_\pm),\ [\gamma]=j \}.
    \end{aligned}$$ In the following for each $U,V\subset M$, we will denote $$\operatorname{dist}(U,V) := \inf_{x\in U,\;y\in V} d(x,y).$$ Choose $\delta>0$. Then there exists an $n\in{{\mathbb{N}}}$ such that for any curve $\gamma\in P_n^\pm$ and $\operatorname{dist}(\gamma(S^1),M\setminus(C_-\cup C_+))<\delta$ it holds that $E(\gamma)\ge A$. We can now say how exactly the finite-dimensional approximation has to be chosen:

-   Choose a $\kappa>0$ large enough such that there exists a homotopy $h:[0,1]\to E^{-1}([0,\kappa))$ in $\Omega$ from $h_0 \in
                P_n^-$ to $h_1 \in P_n^+$ with $$\operatorname{dist}\left(h_t(S^1),M\setminus (C_- \cup
                    C_+)\right) < \delta,
                    \quad \forall t\in[0,1].$$

-   Set $\mathcal{O}:=\{p\in M \,|\, \operatorname{dist}(p,M\setminus(C_-\cup
                C_+))<R\}$, where $R>2\kappa^\frac{1}{2}+\delta$ such that $\mathcal{O}$ contains $\gamma_1,\dots,\gamma_k$.

-   Choose $k$ such that the $(-\nabla E)$-flow of the finite-dimensional approximation respects $C_\pm$, as described 