Union of infinitely often events

Question

I have a simple question. Is the following reasoning correct?

$A_n \subseteq B_n \cup C_n \;\forall\;n \Rightarrow [A_n i.o.] \subseteq [B_n \cup C_n i.o.]$

$\Rightarrow \mathbb{P}([A_n i.o.]) \leq \mathbb{P}([B_n \cup C_n i.o.]) \leq \mathbb{P}([B_n i.o.]) + \mathbb{P} ([C_n i.o.])$?

Hence if the $RHS$ goes to zero $LHS$ also goes to zero?

Thanks in advance.

Answer 1

Yes, it is correct (I assume $X_n\;i.o.$ means $\bigcap_{k\geq 1}\bigcup_{m\geq k} X_m$).

I think it would be advisable, if you would make the additional step

$$ [A_n \;i.o. ] \subseteq [B_n\cup C_n\;i.o] \subseteq [B_n \;i.o. ]\cup [C_n \;i.o. ]. $$

Explanation for the last $\subseteq$: If $x$ appears infinitely often in sets $B_n\cup C_n$, but not infinitely often in $B_n$, it has to be contained infinitely often in $C_n\setminus B_n$, and hence in $C_n$. So $x\in [B_n \;i.o. ]\cup [C_n \;i.o. ]$.