Conjecture: if $a_{n+1}-a_{n}=c $ $\forall n\in\mathbb{N}$ and $b_{n+1}-b_{n} \rightarrow \infty$ then $a_{n}/b_{n} \rightarrow 0$

Question

Is the following proposition true or false?

Given two positive and strictly increasing sequences $a_n$ and $b_n$, if $a_{n+1}-a_{n}=c \in \mathbb{R}$ for every $n\in \mathbb{N}$ and $b_{n+1}-b_{n} \rightarrow \infty$, then $\frac{a_n}{b_n}$ converges to $0$

Answer 1

It holds by Stolz–Cesàro theorem. It suffices that $|a_{n+1}-a_{n}|\leq M$ for some constant $M$.

In fact $b_{n+1}-b_{n} \rightarrow \infty$ implies that $b_n$ is eventually increasing and $$\lim_{n\rightarrow \infty}\frac{a_{n+1}-a_n}{b_{n+1}-b_n}=l=0.$$ Therefore, by Stolz–Cesàro, $$\lim_{n\rightarrow \infty}\frac{a_n}{b_n}=l=0.$$