TITLE: Prove $\lim_{n \to \infty} \frac{\ln(n)}{n}=0$ without L'Hospital's Rule
QUESTION [6 upvotes]: Prove the following without using L'Hospital's Rule, integration or Taylor Series:
$$\lim_{n \to \infty} \frac{\ln(n)}{n}=0 $$
I began by rewriting the expression as: $$\lim_{n \to \infty}{\ln(n^{1/n})} $$
Since the text shows $$\lim_{n \to \infty}{n^{1/n} = 1} $$
I was wondering is the proof just as simple as stating:
$$\lim_{n \to \infty}{\ln(1) = 0} $$
or do I need to apply the squeeze theorem,  use a   $\varepsilon$-N proof, or etc?

REPLY [31 votes]: Since $e^x> x$, we have $\ln x < x$ for all $x  >0$. 
Hence,
$$0 \leqslant \frac{\ln n}{n} = \frac{2 \ln \sqrt{n}}{n} < \frac{2 \sqrt{n}}{n} = \frac{2}{\sqrt{n}} \to 0$$