Suppose $\{X_n,n\ge1\}$ are independent r.v., $E(X_n)=0$, $\operatorname{Var} \left(X_n\right)=\sigma_n^2<\infty$. Set $S_n=\sum_{i=1}^nX_i$ and $s_n^2=\sum_{i=1}^n\sigma_i^2$, assume $$\frac{S_n}{s_n}\overset{d}\to N(0,1)~~~ \text{and}~~~ \frac{\sigma_n}{s_n}\to \rho.$$ Show that $$\frac{X_n}{s_n}\overset{d}\to N\left(0,\rho^2\right).$$

Can we say that since $S_n/s_n=(S_{n-1}+X_n)/s_n\overset{d}\to N(0,1)$ and $S_{n-1}$ is independent with $X_n$, we can conclude that $X_n/s_n$ approaches a normal distribution?