So I was working on a problem and I got stuck. The question is as follows. Let $\pi:E\rightarrow M$ be a vector bundle of rank n with a metric and a metric connection on E. Now suppose that we have a set of curves $\gamma_i:[0,1]\rightarrow M$ such that $\gamma_i(0)=p$ and $\gamma(1)=q$ and let us assume that they converge to a curve $\gamma$.What I want to show is that the associated parallel transports converge.
Here is what I have so far. Let $s_i$ be a sequence of sections over $\gamma_i$ such that $s_i(0)=v$ for all $i$. And so what I did was first selects a local parallel frame $s^i\in\Gamma E$, so that each $s_i$ can be written as
Since these are parallel sections we know that the coefficients are constant. Since the smooth curves converge we know that $s^i\circ\gamma_i$ converges to $s^n\circ\gamma$. What I don't know is how the coefficients are determined relative to the curve so I can use the fact that the curves converge. Any suggestions for what I am not seeing? Also we can assume that the curves are converging in $C^1$ by taking coordinate patches.