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

[Petar Mali]

Homework Statement

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\sqrt{1+\xi^2}-\xi&lt;1

for \xi&gt;0

Homework Equations

The Attempt at a Solution

Is this correct way?

\sqrt{1+\xi^2}-\xi&lt;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&lt;1

for \xi&gt;0

 

[vela]

Yup. That works.

Or you could just reverse your steps.

Let ξ>0. Then 1+ξ²<1+ξ²+2ξ . . .