Every continuous function on a closed nondegenerate rectangle in ${\mathbb{R}}^m$ is Riemann integrable

Riemann's criterion on a nondegenerate rectangle in ${\mathbb{R}}^m$: integrability is equivalent to arbitrarily small Darboux gaps

Lower and upper Darboux sums over a grid partition in ${\mathbb{R}}^m$

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

Open cover, subcover, compact metric space, and compact subset of a metric space

The finite and reverse triangle inequalities for a norm; and for $n\ge 1$ every norm $N$ on ${\mathbb{R}}^n$ satisfies $N(x)\le C\parallel x{\parallel }_1$ and is Lipschitz, hence continuous, for $d_2$

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

The image of a compact metric space under a continuous map is compact, and so is the image of any compact subset

A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value

Lower bound, bounded below, bounded set