According to the differential mean value theorem, | f (x)-f (y) | =| f' (c) (x-y) | < = m | x-y | holds for any x and y.

Xi=i/n，0

Write the integral as the sum of n interval integral values, and f(i/n)/n in the similar sum signal is also written in the form of integral:

F(i/n)/n= integer (from x(i- 1) to x(i))f(i/n)dx.

So we have to prove the left end of the inequality.

= | sum (i= 1 to n) integral (from x(i- 1) to x(i))f(x)dx- sum (i= 1 to n) integral (from x(i- 1)

& lt= sum (i= 1 to n) integral (from x(i- 1) to x(i))|f(x)-f(i/n)|dx

& lt= sum (i= 1 to n) integral (from x(i- 1) to x(i))M(i/n-x)dx.

=M/(2n).