Proposition: Let $p>1$. We have $$ \lim_{n\to +\infty} p^n = +\infty.$$
My professor proved this setting $p=1+\varepsilon$ for some $\varepsilon>0$ and then using Bernoulli's inequality and the comparison theorem. Apart from using the definition of divergence, is this the standard way? I guess my question sort of regards didactics. I'm asking because I feel like the following is more elementary:
$\{p^n\}$ is strictly increasing: $$p^{n+1}>p^n \iff \frac{p^{n+1}}{p^n} =p>1 $$ therefore if it doesn't diverge, it must approach $\alpha\in (1,+\infty)$. But then so does $\{p^{n+1}\}$, which implies $$\lim_{n\to\infty} \frac{p^{n+1}}{p^n}=p=\frac\alpha\alpha=1,$$ contradicting the assumption $p>1$.
I avoided using directly the ratio test as it seems to be more advanced, at least according to my book.