If {${c_n}$}$_{0}^{\infty}$is a sequence of positive numbers increasing to $+\infty$ and if {${a_n}$}$_{0}^{\infty}$ is any sequence of real numbers then
$$\liminf_{n\to\infty} \frac{a_{n+1}-a_n}{c_{n+1}-c_n}\le \liminf_{n\to\infty}\frac{a_n}{c_n}\le \limsup_{n\to\infty}\frac{a_n}{c_n}\le \limsup_{n\to\infty}\frac{a_{n+1}-a_n}{c_{n+1}-c_n}.$$
I am trying to prove this theorem but I am struggling with how to go about doing it.
I am assuming there will be some sort of epsilon approach but I am not exactly sure. Any tips or suggestions to help get me started would be greatly appreciated.
If $S=\limsup\frac{a_{n+1}-a_n}{c_{n+1}-c_n}$ then for any $ε>0$ the number $S+ε$ is an upper bound for almost all of the fractions. Let the exceptions be bounded by $N-1$, then for all $k\ge N$ $$ a_{k+1}-a_k\le(S+ε)(c_{k+1}-c_k) $$ Sum this up from $k=N$ to $n-1$ to get $$ a_n-a_N\le(S+ε)(c_n-c_N)\implies \frac{a_n}{c_n}\le \frac{a_N}{c_n}+(S+ε)(1-\frac{c_N}{c_n}) $$ and for $n\to\infty $ the right side converges to $S+ε$ which makes $S+2ε$ an upper bound for almost all of the fractions $\frac{a_n}{c_n}$.