Root Test For Convergence Question 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 $q&lt1$ then $\frac{1+q}{2}&lt1$ 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.

If $a>1$ then $a=1+h$ for some $h>0$. By the binomial theorem $a^n=(1+h)^n\geq 1+nh$ for $n\geq2$. It follows easily from the last inequality that $a^n\to\infty.$ If $a&lt1$ then $\frac{1}{a}>1$ and the result follows from the case $b>1$. 
