$a$, $b$ are $2$ points in $\mathbb{R}^{2},\rho_{n}(t)\,:\,[0,1]\to\mathbb{R}^{2}$ is a sequence of continuously differentiable constant speed curves with $\|\rho_n'(t)\|=L_n$ for all $t$ from $0$ to $1$ and $\rho_n(0)=a,\,\rho_{n}(1)=b,\forall n$. Suppose that $\lim_{n\to\infty}L_n=\| b-a\|$. I need to show that $\rho_n$ converges uniformly to $\rho(t)=a+t(b-a)$ for $t\in[0,1]$.
So intuitively it's converging to the straight line, for a proof I was thinking about a theorem for uniform convergence of series that says if the series of derivatives converge uniformly to some some function $k$ and the original series converge point wise on some point, then the original series converges uniformly to some $g$ who's derivative is exactly the function $k$. So I think this fits all the criterion except that this is from $\mathbb{R}$ to $\mathbb{R}^{2}$ and I'm not sure if the function extends in this case. ie it's not the derivative here that converges uniformly but rather the arclength so to speak. Is there a similar result for higher dimensions? Thanks!