$X_n\Rightarrow X$, $Y_n\Rightarrow Y$ and $X_n$ & $Y_n$ are independent implies $X$ & $Y$ are independent?

Question

Suppose $X_n$ and $Y_n$ are independent for all $n\geq 1$, and $$X_n\Rightarrow X$$ and $$Y_n\Rightarrow Y$$ as $n\to\infty$, where "$\Rightarrow$" means the convergence in distribution.

Is that right: $X$ and $Y$ are independent? How to prove?

Thank you!