In the comments there are some requests for clarification, you should really respond. As it stands there is not really enough information! Specifically, you did not specify the joint distribution of the $(X_n, Y_n)$. Without some knowledge about that, we cannot answer.
So, first let us assume independence. Then you should be able to prove that $(X,Y)$ also are independent. There could still be some problems. Observe that $x/y =f(x,y)$ and $f$ is not defined for $y=0$ and not continuous in the second argument $y$ in any punctured (at zero) interval around $y=0$. So think about what would happen if $Y$ has probability mass around there.
Without independence: Have a look at Slutsky's theorem, but then you need that either $X$ or $Y$ is a constant (almost surely).