The graph of a continuous function on a closed nondegenerate rectangle in ${\mathbb{R}}^m$ has content zero in ${\mathbb{R}}^{m+1}$

Measure zero and content zero in ${\mathbb{R}}^m$ by countable and finite cube covers

Grid partitions of a rectangle in ${\mathbb{R}}^m$, their cells, refinements and mesh

Axis-parallel rectangles in ${\mathbb{R}}^m$ and their volume

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

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

Continuity of a map between metric spaces, at a point and globally, in the $\epsilon$-$\delta$ form

Each $\parallel \cdot {\parallel }_p$ is a norm on ${\mathbb{R}}^n$, and the induced metrics are exactly $d_1$, $d_2$ and $d_{\mathrm{\infty }}$ of the published metric-spaces page

Integer part: for every real $x$ there is exactly one integer $m$ with $m\le x<m+1$

The canonical natural $\iota (n)=n\cdot 1_F$ of a field

Laws of finite sums and finite products