Let $(X_n)_{n\geq 1}$ be a sequence of random variables such that $\lim_{n \to \infty}Var(X_n)=0$.
Show that if $\lim_{n \to \infty}\Bbb{E}(X_n)=\mu \in \Bbb{R}$ then $X_n \overset{P}{\to} \mu$.
So my teacher solved this problem in class and there is a step which seems reasonable but I can't see how to prove it or how to even be sure it is true.
She says that
$$\{|X_n - \Bbb{E}(X_n)+\Bbb{E}(X_n)-\mu|>\epsilon\}\subset \{|X_n - \Bbb{E}(X_n)|>\frac{\epsilon}{2}\}\cup \{|\Bbb{E}(X_n)-\mu|>\frac{\epsilon}{2}\} $$
And then the proof follows easily. However Im not being able to see how that event inclusion is true, and how could I prove it. How do you prove event inclusions generally?