Questions about Big Oh notation

I'm sure I could think my way through these, but I'm sick and on a tight schedule, so I was hoping someone here could help me out. I would appreciate a verification, with or without proof, of the following assertions:

[tex]
(\mathcal O(\epsilon))^2 = \mathcal O(\epsilon^2)
[/tex]

and

[tex]
\sqrt{1 + \mathcal O(\epsilon^2)} = 1 + \mathcal O(\epsilon^2)
[/tex]

Thanks so much.
 
I think I've managed to show the first one. Suppose [itex]f(\epsilon) = \mathcal O(\epsilon)[/itex] (as [itex]\epsilon \searrow 0[/itex]). Then there exists [itex]C >0, \delta > 0[/itex] such that [itex]0 < \epsilon < \delta[/itex] implies

[tex]
\left| \frac{f(\epsilon)}{\epsilon} \right| \leq C.
[/tex]

To show that [itex](\mathcal O(\epsilon))^2 = \mathcal O(\epsilon^2)[/itex], one simply observes that

[tex]
\left| \frac{f^2(\epsilon)}{\epsilon^2} \right| \leq C^2.
[/tex]
 

mathman

Science Advisor
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The second approximation can be gotten by using the binomial expansion of the left side.
 

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