Let $X_n$ be r.v.'s taking values in $[0, \infty)$. Let $D = \{ X_n = 0 \ \text{for some}\ n \geq 1 \}$ and assume $$P(D \vert X_1, \dots, X_n ) \geq \delta(x) > 0 \quad \text{a.s. on}\ \{X_n \leq x \}.$$ Use Lévy's 0-1 law to conclude that $P(D \cup \{\lim_n X_n = \infty \} ) = 1$.
Let $C=\{\lim_{n \rightarrow \infty}X_n\}$. For this question, I tried to apply the theorem suggested, and I have that $E(1_D|F_N) \rightarrow 1_D$, which is the same as $P(D|X_1,...,X_n)\rightarrow 1_D$. To show $P(C \cup D)$, I'm thinking about showing that $D \subset C^c$. But I'm not show how am I supposed to proceed from here.
On $\{ \liminf_{n \to \infty} X_n \leq M \}$ we have $X_n \leq M + 1$ i.o. so
$$P(D \vert X_1, \dots, X_n) \geq \delta (M + 1) > 0 \quad \text{i.o.}$$
Since the right hand side $\longrightarrow 1_D$, we must have
$$D \supset \{ \liminf_{n \to \infty} X_n \leq M \}.$$
Letting $M \longrightarrow \infty$ implies $D \supset \{ \liminf_{n \to \infty} X_n < \infty \}$ a.s.