HELP how to prove (1-x^2)^n >= 1-nx^2 when x belongs to the interval [-1,1]?

[manuel huant]

i got this question when i read the proof of stone-weierstrass theorem in baby rudin , page 159 , this inequality seems right when n becomes larger, since 1-nx^2 would be negative and (1-x^2)^n always positive, but i don't know how to proved it rigorously using binomial theorem for all n , or is there any other rigorous proof ?

 

[wisvuze]

look up Bernoulli's inequality

 

[micromass]

Try induction.

A more general inequality is called Bernouilli's inequality and states that
$$(1+x)^n\geq 1+nx$$
for $x\geq -1$ and $n\geq 1$. See http://en.wikipedia.org/wiki/Bernoulli's_inequality

 

[manuel huant]

many thanks for your help~