Proving limit of $a^n\to0$ for $|a|<1$ without use of logarithms Prove that $a^n\to0$ as $n\to∞$ for $|a|<1$ without use of logarithms by using properties of the sequence $u_n=|a|^n.$
I've noticed that I should use the subsequence $u_{2n}$, and the fact that $u_{2n}=u_n^2$. However, I don't know where to go from here. I'm not familiar with these types of proofs so a hint/solution would be greatly appreciated.
Thank you!

By ratio test
$$\frac{|a|^{n+1}}{|a|^n}=|a|<1 \implies |a|^n \to 0$$
then since 
$$-|a|^n\le a^n\le |a|^n$$
by squeeze theorem we conclude that
$$a^n \to 0$$
