As Henry Wilton points out this follows from the fact that $\pi_1 N$ is residually finite, i.e. that for every element $g \in \pi_1 N - \{1\}$ there exist a homomorphism $f_g$ to a finite group $G_g$ such that $f_g: \pi_1 N \rightarrow G_g$ is surjective and $f_g(g)\ne 1 \in G_g$.

Thus the kernel of $f_g$ corresponds to a cover where $g$ and its conjugates do not lift. To complete the proof, we invoke the facts that the set of geodesics shorter than a fixed length (here $\epsilon+1$) is finite.

As Henry Wilton points out this follows from the fact that $\pi_1 N$ is residually finite, i.e. that for every element $g \in \pi_1 N - \{1\}$ there exist a homomorphism $f_g$ to a finite group $G_g$ such that $f_g: \pi_1 N \rightarrow G_g$ is surjective and $f_g(g)\ne 1 \in G_g$.

Thus the kernel of $f_g$ corresponds to a cover where $g$ and its conjugates do not lift. To complete the proof, we invoke the fact that the set of geodesics shorter than a fixed length (here $\epsilon+1$) is finite.