$\lim \inf_{n \to \infty} \frac{c_{n+1}}{c_{n}} \le \lim \inf c_{n}^{1/n}$
Theorem 3.37 in Baby Rudin: $\lim\inf\frac{c_{n+1}}{c_n}\leq\lim\inf\sqrt[n]{c_n}\leq\lim\sup\sqrt[n]{c_n}\leq \lim\sup\frac{c_{n+1}}{c_n}$