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## Homework Statement

Prove that

[tex]\frac{1-h}{2}<\sum_{k=1}^{n}x_{2k}(x_{2k+1}-x_{2k-1})<\frac{1+h}{2}[/tex]

where [tex]0=x_1<x_2<\cdots<x_{2n+1}=1[/tex] such that [tex]x_{k+1}-x_{k}<h[/tex] for [tex]1\le k\le 2n[/tex]

## Homework Equations

How to prove? :-)

## The Attempt at a Solution

I need to prove

[tex]\left|-1+2\sum_{k=1}^{n}x_{2k}(x_{2k+1}-x_{2k-1})\right|<h[/tex]

[tex]\left|-1+2\sum_{k=1}^{n}x_{2k}(x_{2k+1}-x_{2k-1})\right|=\left|-1+2\sum_{k=1}^{n}x_{2k}(x_{2k+1}-x_{2k}+x_{2k}-x_{2k-1})\right|\le\left|-1+4h\sum_{k=1}^{n}x_{2k}\right|[/tex]