I'm trying to show that: $$ (X_n,Y_n)\to^p(X,Y)\iff X_n\to^pX,Y_n\to^p Y $$ where $\to^p$ means convergence in probability ($P(||X_n-X||>\varepsilon)\to 0,\forall\varepsilon>0$).
I managed to show $(\Rightarrow)$, but I don't know how to show $(\Leftarrow)$.
Question: How to prove that $$\{||(X_n,Y_n)-(X,Y)||>\varepsilon\}\subseteq\{||X_n-X||>\frac{\varepsilon}{2}\}\cup\{||Y_n-Y||>\frac{\varepsilon}{2}\} $$