$\lim \inf \frac{x_{n+1}}{x_n} \leq \lim \inf \sqrt[n]{x_n}$ for strictly positive real seq $\{x_n\}$
Prove that $\lim \inf \dfrac{x_{n+1}}{x_n}\leq \lim \inf x_n^{1\over n} \leq \lim \sup x_n^{1\over n} \leq \lim \sup \dfrac{x_{n+1}}{x_n}. $