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

[Petar Mali]

Homework Statement

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

for $$\xi>0$$

Homework Equations

The Attempt at a Solution

Is this correct way?

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

suppose

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

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

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

$$0 \geq \xi$$

contradiction

so

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

for $$\xi>0$$

 

[vela]

Yup. That works.

Or you could just reverse your steps.

Let ξ>0. Then 1+ξ²<1+ξ²+2ξ . . .