Let $\gamma:[a,b] \to \mathbb{R}^2$ be a rectifiable curve. Assume we have a sequence of partitions $a = t_{n,1} < ... < t_{n,k_n} = b$
such that $|t_{n,i} - t_{n,i+1}| < \delta_n \forall i \in \{1,...,k_{n}-1\}$.
And a real sequence $(\delta_n)_{n \in \mathbb{N}} \subset \mathbb{R}$ with $\lim_{n \to \infty} \delta_n = 0$.
Show that:
$\lim_{n \to \infty} \sum_{i=1}^{k_n-1}\lVert \gamma(t_{n,i}) - \gamma(t_{n,i+1}) \rVert_{2} = l(\gamma)$, the length of the curve.
We have defined the length of a curve as:
$l(\gamma) = \sup\{\sum_{i=1}^{n-1}|\gamma(t_i) - \gamma(t_{i+1})| \: | a = t_1 < .... < t_n = b\:$ partition of $[a,b] \}$.
Any help would be really appreciated !