If for sequences of random variables $d_n\geq0$ and $r_n>0$ $$ d_n=r_n \left(\frac{1}{n}+\mathcal{O}\left(\frac{1}{n^{1.5}}\right)\right) \text{ as } n\to \infty \,\,Eq.(1)$$ $$\lim\limits_{n\to \infty} Pr\left( r_n \leq 2n\right)=1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,Eq.(2)$$ How can we conclude this? $$\lim\limits_{n\to \infty} Pr\left( d_n \leq 4\right)=1 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,Eq.(3)$$
From Eq(1), if we could conclude $\lim\limits_{n\to \infty} Pr(d_n\leq \frac{r_n}{n})=1$, then we are done. But, I don't know how to do this.