Assuming $M$ is a compact Riemannian manifold. $\gamma_j:[0,1]\rightarrow M$ are equicontinuous curves, namely, there is constant $L$ such that $$ d(\gamma_j(t_1),\gamma_j(t_2))\le L(t_2-t_1) ~~~~~~ \forall t_1\le t_2 \in [0,1],~~\forall \gamma_j\in\{\gamma_j\} $$ And the length of $\gamma_j$ converge, assuming $\lim\limits_{j\rightarrow \infty}\mathcal l(\gamma_j)=h$ (this also means equicontinuous). Then how to show $\{\gamma_j\}$ have uniform convergent subsequence ?
Arzela-Ascoli theorem: Consider a sequence of real-valued continuous functions $\{f_n \}, n \in N $ defined on a closed and bounded interval $[a, b]$ of the real line. If this sequence is uniformly bounded and uniformly equicontinuous, then there exists a subsequence $\{f_{n_k}\},k ∈ N$ that converges uniformly.