Sunday, 28 April 2013

real analysis - Prove limnrightarrowinftyxn=0



I would like to know if my proof is valid, because I did it different from the solution in my textbook (which uses Bernoulli's inequality).



If |x|<1, then limnxn=0.




Proof



For x=0 it is trivial, so we suppose that 0<|x|<1. Let N>log(xε)log(x), then
\begin{align*}
|x|^{n}\end{align*}
implies that limnxn=0 when |x|<1.


Answer



Suppose $\;0


$$x=\frac1r\;,\;\;1

Hint for the last part: assume it is false and use the archimedean property of the reals...


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