If $\{X_n\}$ is a Cauchy seq of r.vs a.s., then why does it converge to some r.v?

Question

I was suddenly suspicious of what title says.

My logics are follows.

$\{X_n\}$ is a Cauchy seq of r.vs $P$-a.s.

$\Leftrightarrow$ $P(\{X_n\}$is a Cauchy $)=1$

$\Leftrightarrow$ $\exists A$ with $P(A)=1$ s.t $\omega \in A \implies \{X_n(\omega)\}$ is a Cauchy seq in $\mathbb{R}$.

Since $\mathbb{R}$ is a complete metric space, $\{X_n(\omega)\}$ has its limit for each $\omega$.

Thus, I define $X$, which is unknown to be a r.v, on $A$ as $X(\omega)= \lim_n X_n(\omega)$ for $\omega \in A$.

Then, X is defined on A but not on whole sample space, say $\Omega$.

At this moment, my questions are

1. No matter how I define $X$ on $\Omega / A$, does $X$ become a r.v?

2. Or my logic above is wrong, and can I find some r.v of being a limit of $\{X_n\}$, in some way?

I appreciate for all your comments in advance.