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 limn→∞xn=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}
implies that limn→∞xn=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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