A question about tail $\sigma $-algebras How do I show formally that the event $\{w\colon\, \lim_{k\rightarrow\infty} X_k(w)$ exists $\}$ is in the tail $\sigma$-algebra of the sequence $X_1, X_2,\ldots$?
Intuitively is quite obvious. The existence of the limit is influenced only by the "last terms", but I don't realize how to write it with precision.
Thank you in advance for the help.
 A: Fix $N\in\mathbb{N}$.
Recall that $X_k(w)$ converges iff it is Cauchy, that is, for every $n\in\mathbb{N}$, there exists some $M\in\mathbb{N}$ such that $|X_i(w)-X_j(w)|=|X_i-X_j|(w)<1/n$ whenever $i,j\geq M$. Also, we can take $M\geq N$, so if $i,j\geq M\geq N$, the function $|X_i-X_j|$ is measurable in $\sigma(X_k,X_{k+1},\ldots)$. To write it formally, we have
\begin{align*}
\left\{w:\lim_{i\to\infty}X_k(w)\text{ exists}\right\}&=\left\{w:(X_k(w))_k\text{ is Cauchy}\right\}\\
&=\left\{w:\forall n\in\mathbb{N},\exists M\geq N,\forall i,j\geq M,|X_i-X_j|(w)<1/n\right\}\\
&=\bigcap_{n\in\mathbb{N}}\bigcup_{M=N}^\infty\bigcap_{i,j=M}^\infty \underbrace{|X_i-X_j|^{-1}(-\infty,1/n)}_{\in\sigma(X_N,X_{N+1},\ldots)}
\end{align*}
Since we have enumerable intersections and unions in $\sigma(X_N,X_{N+1},\ldots)$, then $\left\{w:\lim_{k\to\infty}X_k(w)\text{ exists}\right\}\in \sigma(X_N,X_{N+1},\ldots)$ for every $N\in\mathbb{N}$.
Taking the intersection over every $N$, we conclude that $\left\{w:\lim_{k\to\infty}X_k(w)\text{ exists}\right\}$ is in the tail $\sigma$-algebra.
A: Note 
$$ \{\omega \colon \lim_{k} X_k(\omega)~ \mathrm{exists} \}=\{ \omega\colon \liminf X_k(\omega)=\limsup X_k(\omega)\}$$
and recall that
$$ \liminf X_k, \limsup X_k $$
are $\sigma$-tail measurables.
