while solving a problem John Lee's Introduction to Riemannian manifold exercise 6-17 (exercise itself is not important in this question, however it was the question asking to prove the existence of closed geodesic on non-simply connected compact Riemannian manifold), I've encountered the following problem.
Given a Riemannian manifold $M$ and a sequence of geodesic $g_n : [0,1] \rightarrow M$ (not neccesarily a unit speed) which uniformly converges to a continuous function $g$, is $g$ also a geodesic?
At first glance, I thought that as $g_n$'s satisfy the geodesic differential equation $\ddot{x}^k (t) + \dot{x}^i (t) \dot{x}^j (t) \Gamma^{k}_{ij} (x(t))=0$, using convergence one could easily deduce that $g$ also satisfies the same equation... However, in order to guarantee $g_n'(t) \rightarrow g'(t) $, (based on my analysis knowledge) one should show that the sequence $(g'_n (t))$ uniformly converges, which I could not prove. Then I wasn't able to find a alternative way tackle the problem. Some methods came across my head (to prove local minimality, using universal cover, etc...) but nothing actually worked.
Funny thing (and also quite embarrasing thing) is, that by googling related topics, many people just said this part is "straightfoward". So, I've managed to solve the trickiest part of the exercise, but for whatever reason got blocked by the trivial part. Maybe I'm just having a huge blindspot, so I'm asking for help.
Thank you so much in advance! Have a nice day :)