= Solution
Let $(X_n)$ be <independent random variables> and let
$$
\mathcal T=\bigcap_{n\geq1}\sigma(X_n,X_{n+1},\ldots)
$$
be their <tail sigma-algebra>. The <Kolmogorov zero-one law> states that every $A\in\mathcal T$ has probability zero or one.
Indeed, $A$ is independent of $\sigma(X_1,\ldots,X_n)$ for every $n$. The union of these finite-coordinate sigma-algebras generates $\sigma(X_1,X_2,\ldots)$, and <independence extended from generating pi-systems> shows that $A$ is independent of that entire sigma-algebra. Since $A$ itself belongs to it,
$$
\mathbb P(A)=\mathbb P(A\cap A)=\mathbb P(A)^2,
$$
so $\mathbb P(A)\in\{0,1\}$.
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