# Homework Help: Inequality proof: sqrt(1+xi^2)-xi < 1, for xi > 0

1. Jul 27, 2010

### Petar Mali

1. The problem statement, all variables and given/known data

Show

$$\sqrt{1+\xi^2}-\xi<1$$

for $$\xi>0$$

2. Relevant equations

3. The attempt at a solution

Is this correct way?

$$\sqrt{1+\xi^2}-\xi<1$$

suppose

$$\sqrt{1+\xi^2}-\xi\geq 1$$

$$\sqrt{1+\xi^2}\geq 1+\xi$$

$$1+\xi^2 \geq 1+2\xi+\xi^2$$

$$0 \geq \xi$$

so

$$\sqrt{1+\xi^2}-\xi<1$$

for $$\xi>0$$
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Jul 27, 2010

### vela

Staff Emeritus
Re: Problem

Yup. That works.

Or you could just reverse your steps.

Let ξ>0. Then 1+ξ2<1+ξ2+2ξ . . .