# Proof of an inequality

stanley.st

## Homework Statement

Prove that

$$\frac{1-h}{2}<\sum_{k=1}^{n}x_{2k}(x_{2k+1}-x_{2k-1})<\frac{1+h}{2}$$

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

## Homework Equations

How to prove? :-)

## The Attempt at a Solution

I need to prove
$$\left|-1+2\sum_{k=1}^{n}x_{2k}(x_{2k+1}-x_{2k-1})\right|<h$$

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

## Answers and Replies

boboYO
Draw a diagram. Each term in the sum corresponds to an approximation of the area under the graph in the interval $$(x_{2k-1},x_{2k+1})$$. For each interval, find the difference between
[the approximation] and [the exact value for the area under the graph in the interval $$(x_{2k-1},x_{2k+1})$$]

Add up the differences and you will get your result.

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