$\limsup$ of product of sequences with finite $\limsup$

Question

Assume we have three seqeuences of positive real numbers, say $(a_n)_{n\in \mathbb{N}}$, $(b_n)_{n\in \mathbb{N}}$ and $(c_n)_{n\in \mathbb{N}}$, such that $\displaystyle \limsup \limits _{n\to \infty}\frac{a_n}{b_n}<\infty$ and $\displaystyle \lim \limits _{n\to \infty}\frac{b_n}{c_n}=0$. Do we then have$$\limsup \limits _{n\to \infty}\frac{a_n}{c_n}=\limsup \limits _{n\to \infty}\frac{a_n}{b_n}\frac{b_n}{c_n}=0?$$

Answer 1

Hint: since $\dfrac{a_n}{b_n}$ is bounded, and $\dfrac{b_n}{c_n}\to0$, you have the case in the usual limit "bounded times something tends to $0$ which is 0. Also, since $\dfrac{b_n}{c_n}\to0$ then its limit is finite and $\limsup \dfrac{b_n}{c_n}=\lim\dfrac{b_n}{c_n}$.