Convergence of dyadic tensor composite-Simpson rules

Given: $d\ge 1$ and a continuous $f:[0,1{]}^d\to \mathbb{R}$.

Nested one-dimensional quadrature levels and differences: $Q_L$ is composite Simpson with $2^{L+1}$ equal panels. Regrouping into $2^L$ two-panel blocks gives each block length $H=2^{-L}$ and weights $H(1,4,1)\mathrm{/}6$ at its left endpoint, midpoint, and right endpoint.

Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line: every closed box in Euclidean ${\mathbb{R}}^d$, including $[0,1{]}^d$, is compact.

Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous: a continuous map from a compact metric space to a metric space is uniformly continuous.

Every continuous function on a closed nondegenerate rectangle in ${\mathbb{R}}^m$ is Riemann integrable and Riemann's criterion on a nondegenerate rectangle in ${\mathbb{R}}^m$: integrability is equivalent to arbitrarily small Darboux gaps: a continuous function on the cube is Riemann integrable, and its integral lies between each lower and upper Darboux sum.

Grid partitions of a rectangle in ${\mathbb{R}}^m$, their cells, refinements and mesh: the uniform partition into side-$H$ boxes covers the cube, has disjoint interiors, and its cell volumes sum to one.

Lower and upper Darboux sums over a grid partition in ${\mathbb{R}}^m$ and The lower and upper Darboux integrals over a nondegenerate rectangle in ${\mathbb{R}}^m$: for each cell $C$, the lower and upper contributions are $({\inf }_{C}f)\mathrm{vol}(C)$ and $({\sup }_{C}f)\mathrm{vol}(C)$.

Composite Simpson rule on an even uniform grid: every two-panel Simpson block has coefficients proportional to $1,4,1$, which are nonnegative and sum to $6$.

Laws of finite sums and finite products: finite sums distribute and may be regrouped.

A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value: a continuous function on a nonempty compact metric space has finite maximum and minimum, so its Darboux cell extrema are finite.

For $\mathrm{\mid }r\mathrm{\mid }<1$ the sequence $r^k$ is null, and for $\mathrm{\mid }r\mathrm{\mid }>1$ the sequence $\mathrm{\mid }r{\mathrm{\mid }}^k$ diverges to $+\mathrm{\infty }$ and Limits and Cauchy sequences of reals: for every positive rational $\rho$, all sufficiently large $L$ satisfy $(1\mathrm{/}2{)}^L<\rho$; Laws of integer exponents gives $(1\mathrm{/}2{)}^L=2^{-L}$.

The rationals embed densely in the reals: below every positive real lies a positive rational.

Canonical naturals are positive and strictly increasing and Square roots exist: a unique $\sqrt{a}\ge 0$ with $(\sqrt{a}{)}^2=a$; the positives are $\{x^2:x\mathrm{\ne }0\}$: the canonical natural $d\ge 1$ is positive, so its nonnegative square root $\sqrt{d}$ is nonzero and hence positive; Inverses of positives are positive, and reciprocation reverses order makes $1\mathrm{/}\sqrt{d}$ positive, and Sign rules for products and monotonicity of multiplication makes $\delta \mathrm{/}\sqrt{d}$ positive.