Suppose that $f: [a,b] \rightarrow \mathrm{R}^n$ is continuous with a derivative f' whose norm is Riemann-integrable. To demonstrate the arclength integral formula, I'm trying to prove that, for every $n \in \mathrm N$, there exists a tagged partition $(P_n,\xi)$ of $[a,b]$ such that
|P_n| < \frac 1n\text{ and }|f(t_i) - f(t_{i-1}) - f'(\xi_i)(t_i - t_{i-1})| < \frac{t_i - t_{i-1}}n
for every interval $[t_{i-1},t_i]$ of the partition, where $\xi_i$ is the corresponding tag of the interval.
If f' is continuous, the statement is a consequence of the uniform differentiability of $f$. However, I don't think this will be useful in proving the general case. Any suggestions on how this can be proved?