Deducing a.s. finiteness from a.s. finiteness of conditioned variable

Deducing a.s. finiteness from a.s. finiteness of conditioned variable

Consider a sequence of non-negative random variables $X_n$ and the natural
filtration $\mathcal{G}=\sigma(X_1,X_2,...)$
Suppose that $\lim_{n\rightarrow \infty} X_n$ exists and is finite
$\mathbb{P}(\cdot \vert \mathcal{G})-a.s.$.
Why is it true that then also $\lim_{n\rightarrow \infty} X_n$ exists and
is finite $\mathbb{P}-a.s.?$