I'm reading about convergence of Riemannian manifolds:
A sequence of pointed complete Riemannian manifolds is said to converge in the pointed $C^{m, \alpha}$ topology $(M_i, p_i, g_i) \rightarrow (M,p,g)$ if for every $R > 0$ we can find a domain $\Omega \supset B (p,R) \subset M$ and embeddings $F_i : \Omega → M_i$ for large i such that $F_i (\Omega) \supset B(p_i ,R)$ and $F_i^* g_i \rightarrow g$ on $\Omega$ in the $C^{m, \alpha}$ topology.
Then it is stated that it is easy to see that this type of convergence implies pointed Gromov-Hausdorff convergence.
I haven't been able to figure out how to see this. Can you help me with this? I guess we need to contruct some metric on $M \sqcup M_i$ using the $C^{m, \alpha}$-convergence of $F^*g_i$ to see this?