Sei \( f_k = f(x_k^{(n)}) \) und \( s(x) = f_k + \frac{f_{k+1}-f_k}{x_{k+1}-x_k}(x-x_k) \) dann gilt $$ \left| T(f,n) - \int_a^b f(x) dx \right| \le \\\sum_{k=0}^{n-1} \left| \int_{x_k}^{x_{k+1}} \left[ s(x) - f(x) \right] dx\right| \le \sum_{k=0}^{n-1} \left( \sup_{y \in [x_k, x_{k+1}]} |f''(y)| \right) (x_{k+1}-x_k)^3 \\h^2 \left( \sup_{y \in [a,b]} | f''(y) | \right) \sum_{k=0}^{n-1}(x_{k+1}-x_k) = h^2 \left( \sup_{y \in [a,b]} | f''(y) | \right) (b-a) $$
