= Finite bad-cell estimate for Darboux sums
Let $|f|\leq M$ on $[0,1]$ and let $P$ have $r$ interior division points. At most $r$ cells of the equal-length partition $D_n$ have one of those points in their interior. Their total length is at most $r/n$, and their oscillation is at most $2M$. Every other cell lies in a single cell of $P$. Therefore
$$
U(f,D_n)-L(f,D_n)\leq U(f,P)-L(f,P)+\frac{2Mr}{n}.
$$
This comparison of <Darboux sums> avoids the incorrect assumption that $D_n$ refines $P$ for all large $n$.
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