While reading the proof of the root test for convergence in my notebook I came across this two claims:
let $\{a_n\}$ be a positive sequence and $\lim \limits_{n\to \infty}\sqrt[n]{a_n}=q. $
claim (1): if q<1 then \frac{1+q}{2}<1 therefore $\lim \limits_{n\to \infty}(\frac{1+q}{2})^n=0$ - how do I prove that the sequence converges to $0$?
claim (2): if $q>1$ then $\frac{1+q}{2}>1$ therefore $\lim \limits_{n\to \infty}(\frac{1+q}{2})^n=\infty$ - how do I prove that the sequence converges to $\infty$?
Thanks a lot.