Is $\lim\limits_{n\rightarrow \infty}f(c_n)=\infty \Rightarrow \lim\limits_{n\rightarrow \infty}\frac{f(c_n)}{n}\neq 0$ true?

Question

Let $(c_n)$ be a sequance with $\lim\limits_{n\rightarrow \infty} c_n\rightarrow c$ and $f(x)$ a function that convegres to infinity for $\lim\limits_{n\rightarrow \infty}f(c_n)=\infty$.

I want to proof that every infinite series of the form$\lim\limits_{n\rightarrow \infty} \sum \limits_{k=0}^{n}\frac{f(c_k)}{k}$ diverges in this case because $a_k=\frac{f(c_k)}{k}$ is not a zero sequence.

Is my assumption $$\lim\limits_{n\rightarrow \infty}f(c_n)=\infty \Rightarrow \lim\limits_{n\rightarrow \infty}\frac{f(c_n)}{n}\neq 0$$ correct?

Thoughts: My gut feeling tells me that it is not so easy, because $\lim\limits_{n\rightarrow \infty}\frac{f(c_n)}{n}\neq 0$ would only be true, if $f(c_n)$ has a higher rate of convergence in comparison to $n$ but I'm not sure.

Answer 1

No, $f(c_n) = \sqrt{n}$ is a counterexample. $\lim_{n \rightarrow \infty} \sqrt{n} = \infty$, but $\lim_{n \rightarrow \infty} \sqrt{n}/n = 0$