Let $a$ and $b$ be two points of $\mathbb{R}^2$.Let $σ_n : [0, 1] → \mathbb{R}^2$ be a sequence of continuously differentiable constant speed curves with $||σ_n'(t)|| = L_n$ for all $t ∈ [0, 1]$ and $σ_n(0) = a$ and $σ_n(1) = b$ for all $n$. Suppose that $\lim_{n→∞} L_n = ||b − a||$. Show that $σ_n$ converges uniformly to $σ$, where $σ(t) = a + t(b − a)$ for $t ∈ [0, 1]$.
My Try:
Intuitively this is clear since it talks about a sequence of paths from $a$ to $b$ that converges to the straight line through $a$ and $b$.
For a rigorous proof, I wanted to show first that $σ_n'(t)$ converges uniformly. But all I have is given $\epsilon>0$ there is $N$ such that $| \;||σ_n'(t)||-||b − a||\;|<\epsilon$ for all $n>N$, which does not support what I need to prove. Any suggestion please..
Main ideas: 1. Since $\sigma_n'$ is bounded, $\{\sigma_n\}$ is equicontinuous. The endpoint condition and Arzela-Ascoli shows then that there is a subsequence $\sigma_{n_k}$ that converges uniformly to some $\sigma_0(t)$ on $[0,1].$
$$L_{n_k} \ge |\sigma_{n_k}(t) -a| + |b - \sigma_{n_k}(t)| \to |\sigma_{0}(t) -a| + |b - \sigma_{0}(t)| > |b-a|$$
for all $k,$contradiction.
Thus we can write $\sigma_0(t) = a + f(t)(b-a)$ for some $f:[0,1] \to [0,1].$ Now use the fact that $|\sigma_{n_k}(t) - a| = tL_{n_k},$ for all $k,t. $ This will give $f(t) = t.$ Thus $\sigma_0$ has the desired form.
Verify that it's enough to have proved this for a subsequence.