Question

Can somebody help me prove the statement below?

Answer 1

Prove that the succession \((S_n)_{n=1}^{\infty}\) given by \[S_n=\sum _{i=1} ^ n M_i \Delta x: M_i=\max \left\{ f(x):x \in \left[ x_{i-1},x_i \right] \right\} , \Delta x=\frac{ x_n-x_0 }{ n } \]Is monotonic and has a lower limit.