Kolmogorov's Zero-One law

Let ($\Omega, \mathcal{F}, P$) be a probability space, and let $\xi_1, \xi_2, ...$ be a sequence of random variable. Let $\mathcal{F}^{\infty}_n = \sigma(\xi_n, \xi_{n+1}, ...)$ be the $\sigma$-algebra generated by $\xi_n, \xi_{n+1}, ...$, and write$$\begin{equation*}\mathcal{T} = \bigcap^{\infty}_{n=1} \mathcal{F}^{\infty}_n .\end{equation*}$$Since the intersection of $\sigma$-algebras is again a $\sigma$-algebra, $\mathcal{T}$ is a $\sigma$-algebra. It is called a ---tail algebra---, because every event $A \in \mathcal{T}$ is independent of the values of $\xi_1, ...,\xi_n$ for every finite number $n$, and is determined, so to speak, only by the behavior of the infinitely remote values of $\xi_1,\xi_2, ....$. \\Since, for every $k \geq 1$,$$\begin{equation*}A_1 = \left\{ \sum^{\infty}_{n=1} \frac{\xi_n}{n} \, \text{converges} \right\} = \left\{ \sum^{\infty}_{n=k} \frac{\xi_n}{n} \, \text{converges} \right\} \in \mathcal{F}^{\infty}_k\end{equation*}$$we have $A_1 \in \bigcap_k \mathcal{F}^{\infty}_k \equiv \mathcal{T}$. In the same way, if $\xi_1,\xi_2, ....$ is any sequence,$$\begin{equation*}A_2 = \left\{ \sum^{\infty}_{n=1} \xi_n \, \text{converges} \right\} \in \mathcal{T}\end{equation*}$$