Show that $X_n \overset{p}{\to} X$ and $Y_n \overset{p}{\to} Y$ implies $(X_n, Y_n) \overset{p}{\to} (X,Y)$. How can I show that if $X_n \overset{p}{\to} X$ and $Y_n \overset{p}{\to} Y$, then $(X_n, Y_n) \overset{p}{\to} (X,Y)$? that is, if $X_n$ converges in probability to $X$, and $Y_n$ converges in probability to $Y$, then we have convergence in the joint sense. I've tried to use an epsilon approach, but cannot seem to get a bound. Does anyone have any hints? Thanks!

For any $\epsilon>0$

$$P\left\\{d\left((X_n,Y_n),(X,Y)\right)>\epsilon\right\\}\le P\\{|X_n-X|>\epsilon/\sqrt{2}\\}+P\\{|Y_n-Y|>\epsilon/\sqrt{2}\\}\to 0.$$

Here, $d(z, w)=\|z-w\|_2$.