I was looking over a proof that every rectifiable curve is parametrizable by its arc length and found that it was used that the arc length is additive.
It is pretty obvious that this is true, but how can you show it rigorously?
For a curve $c: \mathbb R \supset [a,b] \rightarrow (X,d)$ in a metric space $(X,d)$ the arc length is defined as $$L(c_{|_{[a,b]}})=\sup \left\{\sum _{i=1}^{n}d(c(t_{i}),c(t_{i-1})):n\in \mathbb {N} {\text{ and }}a=t_{0}<t_{1}<\dotsb <t_{n}=b\right\}.$$
For any $u < v < w \in [a,b]$ how can I show that $L(c_{|_{[u,w]}})=L(c_{|_{[u,v]}}) + L(c_{|_{[v,w]}})$ ?
There exists some $t^*$ such that $c(t^*) = v$
For any partition $t_0<t_1<\cdots<t_k\le t^*<t_{k+1}\cdots<t_n$ $L(c_{|u,w|}) = \sup \{\sum_\limits {i=1}^k d(c(t_i), c(t_{i-1})+\sum_\limits {i=k+1}^n d(c(t_i), c(t_{i-1})\} = L(c_{|u,v|}) + L(c_{|v,w|})$