If this sequence of paths tend to a certain path $\gamma$, does the integrals along those paths tend to the integral along $\gamma$?
Solution 1:
Suppose $\{ \gamma_n \}$ converges uniformly to $\gamma_0$. Suppose also that each $\gamma_n$ is $C^1$ and its derivative converges uniformly to some function $g$ and $f$ is continuous. It is a classical result then that $\gamma_0$ is differentiable and its derivative is $\gamma_0’ = g$
Then $f$ is bounded on the trace of $\gamma_n$ and $f(\gamma_n) \gamma_n’ $ converges uniformly to $f(\gamma_0) \gamma_0’$. Therefore $\int^z_0 f(\gamma_n) \gamma_n’$ converges uniformly to $\int^z_0 f(\gamma_0) \gamma_0’ $, which is the result you are looking for.