Show that $\lim_{n\to\infty} a^{\frac1n}=1$

Question

Any ideas about how to show this neatly? I have seen a way using inequality between geometric and arithmetic mean. Is there a simpler way?

Answer 1

Assume that $a>1$. Write $a=1+p$ for some $p>0$. We know the inequality that $(1+p/n)^{n}\geq 1+p$, so $a^{1/n}\leq 1+p/n$. Now $a^{1/n}\geq 1$, squeeze theorem gives the result. For $a=1$, nothing to prove. For $a<1$, consider $1/a^{1/n}$.