Hello everybody, I am new in the forum, nice to meet you, please help me with this problem.

Let such that: for all , i.e. is an isometry, for all . Show that , for all . Conclude that there is and a such that .

Thanks (Rofl)(Rofl)(Rofl)

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- Feb 1st 2010, 06:58 PMmiguematedifferentiable isometry
Hello everybody, I am new in the forum, nice to meet you, please help me with this problem.

Let such that: for all , i.e. is an isometry, for all . Show that , for all . Conclude that there is and a such that .

Thanks (Rofl)(Rofl)(Rofl) - Feb 1st 2010, 06:59 PMDrexel28
- Feb 1st 2010, 07:13 PMmiguemate
Hello Drexel, thanks for your answer, here is my work

Using Cauchy’s inequality

.

By hypotesis, , then

(1)

Taking by

,

I get .

By the mean value theorem, there is a c in <0,1> such that

This latter using (1)

Hence

.

That's all, but i do not what else to do

Hugs - Feb 1st 2010, 07:14 PMmiguemate
Hello Drexwl, thanks for your answer, here is my work

Using Cauchy’s inequality

.

By hypotesis, , then

(1)

Taking by

,

I get .

By the mean value theorem, there is a c in <0,1> such that

This latter using (1)

Hence

.

That's all, but i do not what else to do

Hugs - Feb 9th 2010, 09:28 PMmiguemate
any idea?