 Ratio-one interpolation for shrinking-window occupation integrals

For any measurable real path $x$, put $A_n(t)=n{\int }_0^t{1}_{\{|x(s)|<1/n\}}ds$. If $n_k\le n\le n_{k+1}$, nested windows give $(n/n_{k+1})A_{n_{k+1}}(t)\le A_n(t)\le (n/n_k)A_{n_k}(t)$. Consequently, when $n_{k+1}/n_k\to 1$ and the subsequence $A_{n_k}$ converges uniformly on bounded time intervals, the full sequence does also, to the same limit. This deterministic interpolation turns summable-subsequence First Borel-Cantelli lemma estimates into almost-sure convergence without incorrectly claiming that convergence in $L^2$ alone implies full-sequence almost-sure convergence.