triangle inequality dividing e and seperating probabilities

Question

I understand the first use of triangle inequality but I dont understand the last part where they divided e by 2 and put a greater than or equal too sign? why are we allowed to do that?

triangle inequality

Answer 1

$$ \left| X_n-X\right| + \left| Y_n-Y\right|> \epsilon\implies \cases{either\ &$\left| X_n-X\right|>\frac{\epsilon}{2}$\\ or& $\left| Y_n-Y\right|> \frac{\epsilon}{2}\ $, } $$ because if neither of those two latter inequalities hold then neither can the first one either.

Thus,

$\ \left\{\left| X_n-X\right| + \left| Y_n-Y\right|> \epsilon\right\}\\ \hspace{5em}\subseteq\left\{\left| X_n-X\right|>\frac{\epsilon}{2}\right\}\cup\left\{\left| Y_n-Y\right|>\frac{\epsilon}{2}\right\}\ ,$

and therefore \begin{align} P\left(\left| X_n-X\right| +\right.&\left. \left| Y_n-Y\right|> \epsilon\right)\\ &\le P\left(\left[\left| X_n-X\right|>\frac{\epsilon}{2}\right]\vee\left[\left| Y_n-Y\right|>\frac{\epsilon}{2}\right]\right)\\ &\le P\left(\left| X_n-X\right|>\frac{\epsilon}{2}\right)+P\left(\left|Y_n-Y\right|>\frac{\epsilon}{2}\right)\ . \end{align}