= Ratio-one interpolation for shrinking-window occupation integrals
For any measurable real path $x$, put $A_n(t)=n\int_0^t\mathbf1_{\{|x(s)|<1/n\}}ds$. 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 $n_{k+1}/n_k\to1$ 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.
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