Asymptotic equivalence and Big Oh notation

Question

Let $a_n, b_n$ be two positive real sequences. Suppose that $\lim_{n\to\infty}\frac{a_n}{b_n}=1$ and $b_n\leq \frac{c}{n}$ for some $c>0$ and all sufficiently large $n$. Does $a_n\leq \frac{C}{n}$ for some $C>0$ and all $n$ large enough?

Answer 1

Hint. Let $\alpha_n=\frac{a_n}{b_n}$. Then, $a_n=b_n\cdot\alpha_n$. Now, note that there is a $N$ such that for all $n>N$ we have $\frac{1}{2}<\alpha_n<\frac{3}{2}$.