Why does (x²ᵢ₊₁ + xᵢ₊₁xᵢ + x²ᵢ) < 3δ in a Riemann integral proof?

[Bashyboy]

Homework Statement

Here is a link to the problem which I am currently working on:

http://math.umn.edu/~jara0025/Math4603/Math4603H9Answers.pdf

Homework Equations

The Attempt at a Solution

[/B]
The specific problem I am working on is found on page 3, and is the first problem on that page. The step I am having trouble with is

$$= \sum_{i=0}^{n-1} (x_{i+1} - x_i)(x^2_{i+1} + x_{i+1}x_i + x_i^2) (x_{i+1} - x_i)$$

$$ < \delta 3 \sum_{i=0}^{n-1} (x_{i+1} - x_i)$$

What is going on?

 

[Dick]

Homework Statement

Here is a link to the problem which I am currently working on:

http://math.umn.edu/~jara0025/Math4603/Math4603H9Answers.pdf

Homework Equations

The Attempt at a Solution

[/B]
The specific problem I am working on is found on page 3, and is the first problem on that page. The step I am having trouble with is

$$= \sum_{i=0}^{n-1} (x_{i+1} - x_i)(x^2_{i+1} + x_{i+1}x_i + x_i^2) (x_{i+1} - x_i)$$

$$ < \delta 3 \sum_{i=0}^{n-1} (x_{i+1} - x_i)$$

What is going on?

The point is that $x_i \le 1$ and $(x_{i+1}-x_i) \lt \delta$ for all $i$. Try using those inequalities.