I want to show that $\left[\begin{array}{c}X_{n} \\ Y_{n}\end{array}\right] \rightarrow^{D} W$ where $W$ has a bi-variate normal distribution with mean vector 0 and covariance matrix $\left[\begin{array}{cc}1+\alpha^{2} & \alpha \\ \alpha & 1\end{array}\right]$.
Here, for $n=1,2, \ldots$, $X_{n}=Z_{1}+\alpha_{n} Z_{2}$ and $Y_{n}=Z_{2}$, where $Z_{1}$ and $Z_{2}$ are independent standard normal random variables and, $\alpha_{1}, \alpha_{2}, \ldots$ is a sequence of real numbers convergent to $\alpha$ .