= Exceedance equivalence for increasing thresholds
{title2=$\limsup_n\{\max_{k\leq n}X_k>a_n\}=\limsup_n\{X_n>a_n\}$}
Let $(a_n)$ be a nondecreasing real <sequence> tending to $+\infty$, and let $(x_n)$ be any real <sequence>. The running maximum exceeds $a_n$ infinitely often exactly when $x_n>a_n$ infinitely often. Applying this pathwise to <random variables> gives the displayed <event> equality, without any <independence> assumption.
One implication follows from $x_n\leq\max_{k\leq n}x_k$. For the other, if $x_n\leq a_n$ from some $N$ onwards, every later value with $N\leq k\leq n$ satisfies $x_k\leq a_k\leq a_n$. The finite prefix also lies below $a_n$ for all sufficiently large $n$, because $a_n\to\infty$. Therefore the entire running maximum eventually stays below the threshold.
Divergence of the thresholds prevents one large early observation from creating infinitely many crossings. Monotonicity ensures that a later running-maximum crossing is controlled by the corresponding earlier individual threshold. These two conditions explain why the resulting <event> belongs to the <tail sigma-algebra> when the thresholds are deterministic.
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