Suppose I have a connection $\nabla$ in a riemannian manifold $M$. Let $X_n\rightarrow X$ be smooth vector fields converging in $C^{\infty}$. I was wondering if $\nabla _{X_n}Y\rightarrow \nabla_{X}Y$?
I think this is true , if we look in local coordinates in an open set that is contained in a compact set , we can bound the christoffel symbols and the terms appearing for $Y$ and then using the convergence I think we get the desired result.
However I wanted to make sure this was correct or If I am making some mistake.
Any insight is appreciated, thanks in advance.