Let $(\Omega, \mathcal{F},P)$ be a probability space and let $(X_n)_{n\geq 0}$ be a real-valued stationary process. Let $B\subset \mathcal{R}$, then why is $$ A:=\{X_n \in B,\text{ i.o.}\} $$ invariant? Moreover, why is $$ \tilde A:=\{X_n \geq 2^n,\text{ i.o.}\} $$ not invariant?
My reasoning can be presented as follows. Let $\mathcal{I}_X$ be the invariant $\sigma$-field. Then, it suffices to show that $A^c \in \mathcal{I}_X$. Notice that $$ A^c = \{\omega: \exists N\,\text{s.t.}\, X_n(\omega) \in B\,\forall n\geq N\}. $$ Hence, $\{X_n \in B,\text{ i.o.}\}^c = \{X_{n+1} \in B,\text{ i.o.}\}^c$, which shows that $A^c \in \mathcal{I}_X$. Why the same reasoning does not apply to $\tilde A$?