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).