Suppose that $\frac{p_n}{(\lambda_n \rho_n)^k}=o(\rho_n ^k)$, where $k$ is a positive integer, and the condition about $\rho_n$ is that $M_3n^{-c_1/2} \leq \rho_n \leq M_1$, where $0 \leq c_1 \leq 1/2$, $M_1$ and $M_3$ are positive constants. Then how can I derive $\frac{p_n}{(\lambda_n \rho_n)^k} \rightarrow 0$, as $n \rightarrow \infty$?
My thought is as follow
Since $\frac{p_n}{(\lambda_n \rho_n)^k}=o(\rho_n ^k)$, $\lim_{n \rightarrow \infty}\frac{p_n}{(\lambda_n \rho_n)^k \rho_n^k}=0 $.
Since $\rho_n \leq M_1$, $\rho_n^k \leq M_1^k$. Then $\lim_{n \rightarrow \infty}\frac{p_n}{(\lambda_n \rho_n)^k M_1^k}=0 $.
I derive that $\frac{p_n}{(\lambda_n \rho_n)^k}=o(M_1^k)=M_1 ^ko(1) \rightarrow 0$.
However, my supervisor said it is wrong. How can I derive $\frac{p_n}{(\lambda_n \rho_n)^k} \rightarrow 0$, as $n \rightarrow \infty$?