= Solution
The <tail sigma-algebra> of this <sequence> is
$$
\mathcal T=\bigcap_{r\geq1}\sigma(X_r,X_{r+1},\ldots).
$$
For each fixed $r$, the preceding <event> equality gives
$$
A=\bigcap_{N\geq r}\bigcup_{n\geq N}\{X_n>a_n\}.
$$
Every <event> on the right is determined by coordinates with index at least $r$. Therefore $A\in\sigma(X_r,X_{r+1},\ldots)$ for every $r$, and $A\in\mathcal T$. The <random variables> $X_n$ are <independent>, so <Kolmogorov's zero–one law> applies: every <tail event> has <probability> zero or one. Hence \b[$\mathbb P(A)\in\{0,1\}$].
The role of <independence> can also be seen directly. A <tail event> is independent of each finite initial <sigma-algebra> $\sigma(X_1,\ldots,X_m)$. The <Monotone class theorem> extends this <independence> to $\sigma(X_1,X_2,\ldots)$; since $A$ itself lies there, it is independent of itself. Thus $\mathbb P(A)=\mathbb P(A)^2$.<Probability: Theory and Examples (fifth edition) by Rick Durrett>[§2.5, Theorem 2.5.3, printed pp. 81–82: the tail sigma-field zero–one law]{ref}
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