For compact subsets of ${\mathbb{R}}^m$, measure zero and content zero coincide

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

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

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

A subset of a metric space is open in the subspace metric exactly when it is the trace of an open set of the ambient space, and it is compact as a metric space in its own right exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it

The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement

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

For $\mathrm{\mid }r\mathrm{\mid }<1$, ${\sum }_{k\ge 0}{r}^k=1\mathrm{/}(1-r)$, and for $\mathrm{\mid }r\mathrm{\mid }\ge 1$ the series diverges

Finite sums and finite products, by recursion

Laws of finite sums and finite products