Let $f:[0,\infty)\to\mathbb{R}^n$ be a Lipschitz function satisfying $\int_0^\infty\|f(t)\|dt < \infty$ and $0 = t_0 < t_1 < t_2 < \dots$ be a divergent sequence satisfying $\sup_{i\in\mathbb{N}}(t_i-t_{i-1})<\infty$. I am trying to prove or disprove the following:
there exists $M\in(0,\infty)$ such that $\sum_{i=1}^\infty\int_{t_{i-1}}^{t_i}\|f(t) - f(t_{i-1})\|dt \le M\sup_{i\in\mathbb{N}}(t_i-t_{i-1})$
but cannot make any progress. Would you give me any hint for this problem?
As is, I think that your integral $I$ is not ensured to be convergent.
Take $f$ to be piecewise affine, with $f(0)=0$ and for all $n \geq 10$, $$f(n)=\frac{1}{n\ln{n}},$$ $$f\left(n \pm \frac{1}{\ln{n}}\right)=0.$$
Take $t_n=n+10$.
Then $$\infty=\sum_n{f(t_{n-1})(t_n-t_{n-1})} \leq \sum_n{\int_{t_{n-1}}^{t_n}{\|f-f(t_{n-1})\| +\|f\|}} \leq I + \int_{\mathbb{R}^+}{\|f\|}.$$