energy $\kappa_0$ in $K$. By definition of $A$ and by using \[it:existsU\]) (when $d_j$ is homologically visible), there exist disjoint neighborhoods $U_j$ of the $S^1\cdot d_j$’s such that $U_j \cap E^{-1}([0,\kappa_0))$ is connected for all $j$. Since $\partial C_\pm$ are not totally geodesic, we know that the $d_j$’s are not contained in $\partial K$ and we can assume that $U_j \subset K$. Now because there are only finitely many closed geodesics in $K\cap E^{-1}([0,\kappa_0+\varepsilon])$ for $\varepsilon>0$ small enough, one can take $\varepsilon>0$ such that there is no closed geodesic in $K \cap E^{-1}((\kappa_0,\kappa_0+\varepsilon])$. By the definition of $\kappa_0$ there exists a homotopy $\beta\in\Pi$ satisfying $E(\beta_t)\le \kappa_0+\varepsilon$ for all $t\in[0,1]$ such that $\beta_t\in K$. Choose neighborhoods $V_j$ of $S^1\cdot d_j$ such that $\overline{V_j}\subset
\text{int}(U_j)$ and use property \[it:existsT\]) on the neighborhood $V:=\bigcup_{k=1}^l V_j$ of closed geodesics of energy $\kappa_0$ in $K$ to obtain a $\tau>0$ with the property that for the homotopy $\phi_\tau\circ\beta$ we have that $(\phi_\tau\circ\beta)_t \in K$ implies $E((\phi_\tau\circ\beta)_t)<\kappa_0$ or $(\phi_\tau\circ\beta)_t \in V$. Now $(\phi_\tau\circ\beta)^{-1}(V) = \bigcup_{r=1}^m
(t_r,t'_r)$ and by our choice of the $V_j$ we have $(\phi_\tau\circ\beta)([t_r,t'_r])\subset U_j$ and for the endpoints $(\phi_\tau\circ\beta)_{t_r},(\phi_\tau\circ\beta)_{t'_r}\in U_j \cap
E^{-1}([0,\kappa_0))$ for some $j\in\{1,\ldots,l\}$ (which is why we applied \[it:existsT\]) only to $V$ 