= Solution
For independent random variables $X_1,X_2,\ldots$, let $\mathcal F_n=\sigma(X_1,\ldots,X_n)$, $\mathcal F=\sigma(X_1,X_2,\ldots)$, and let the <tail sigma-algebra> be $\mathcal T=\bigcap_n\sigma(X_{n+1},X_{n+2},\ldots)$. <Kolmogorov zero-one law> states that every $A\in\mathcal T$ has probability zero or one.
Such an event is independent of each $\mathcal F_n$ by independence of the variables. The collection of events $B\in\mathcal F$ satisfying $\mathbb P(A\cap B)=\mathbb P(A)\mathbb P(B)$ is a <Dynkin system>: it contains the whole space and is closed under complements and disjoint countable unions. It contains the <pi-system> $\bigcup_n\mathcal F_n$, which generates $\mathcal F$. The <pi-lambda theorem> therefore makes $A$ independent of all of $\mathcal F$. Since $A$ itself belongs to $\mathcal F$,
$$
\mathbb P(A)=\mathbb P(A\cap A)=\mathbb P(A)^2,\qquad\boxed{\mathbb P(A)\in\{0,1\}.}
$$
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