notation, let us identify the image of the loop $c$ to $S^1\times \{ 0\}$, with $c(s)=(s,0)$ for $s\in S^1$, so that $M\setminus c(S^1)$ is the disjoint union of the neighborhood $S^1\times (-\infty,0)$ of $-\infty$ and the neighborhood $S^1\times (0,+\infty)$ of $+\infty$ (we only need this identification to be a homeomorphism). We now use Lemma \[lem:delta\] to find $n\leq \lceil{1/k(\delta-\varepsilon)}\rceil +1$ geodesic chords $\alpha_1,\ldots,\alpha_n$ lying inside $S^1\times [0,+\infty)$ and intersecting $c$ only at endpoints so that the unbounded component $C_+$ of $S^1\times (0,+\infty) \setminus \bigcup_i \alpha_i([0,1])$ is locally convex and not a closed geodesic. We can do the same on the other side and eventually find two locally convex neighborhoods $C_\pm$ of $\pm\infty$ whose boundaries are not totally geodesic and with non-intersecting closure. To make the statement precise, let $p : S^1 \to [0,1)$ be the natural bijection of $S^1={{\mathbb{R}}}/{{\mathbb{Z}}}$ to the fundamental domain $[0,1)$ and $\pi : M\simeq S^1\times{{\mathbb{R}}}\to [0,1)$ be the induced projection so that $\pi\circ c(t) = t$ for $t\in[0,1)$.

Let $k\in{{\mathbb{N}}}^*$ and $\delta>0$ be given by Lemma \[lem:delta\]. Let $\varepsilon < \frac{\delta}{4}$ be a small non-zero real number. Up to a translation of the parametrisation of $c$, we construct inductively a family $\alpha_1,\dotsc,\alpha_n$, $n\leq \lceil{1/k(\delta-\varepsilon)}\rceil$, of geodesic 