The attached document (page 17 - remark 3.4.2) fuente shows a formula that is derived from the mean value theorem. However, as much as I have searched for reference to said formula, I cannot find how to connect it with said theorem. I know that the "inequality of the mean value theorem" exists, but its formula is different from the one applied in the document.
Where $$\left\|∏\right\| = \max_{(j=0,\ldots, n-1)}(t_{j+1}-t_j)$$
$$\sum_{j=1}^{n-1} [f'(t^*_j)]^2 (t_{j+1}-t_j)^2 \le \left\|∏\right\|\sum\limits_{j=1}^{n-1} [f'(t^*_j)]^2 (t_{j+1}-t_j) $$
What is the process to arrive at this inequality? Thank you.
Fuentes uses the mean value theorem only to show that the equality $$ \sum_{j=0}^{n-1}[f(t_{j+1})-f(t_j)]^2=\sum_{j=0}^{n-1}[f'(t^*_j)]^2(t_{j+1}-t_j)^2 $$ holds. The inequality $$ \sum_{j=0}^{n-1}[f'(t^*_j)]^2(t_{j+1}-t_j)^2\le \underbrace{\max_{j=0,...,n-1}(t_{j+1}-t_j)}_{||\Pi||}\sum_{j=0}^{n-1}[f'(t^*_j)]^2(t_{j+1}-t_j) $$ is trivial because for every $j$ you clearly have $t_{j+1}-t_j\le \max\limits_{j=0,...,n-1}(t_{j+1}-t_j)\,.$