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Cousin's lemma yields the Heine–Borel theorem on a compact interval

Example

Cousin's lemma implies that every open cover of a compact interval $[a, b]$ has a finite subcover, the interval form of Heine-Borel.

Facts & Assumptions

Given: An open cover $U$ of $[a, b]$ .

[L1]

Every gauge on a compact interval admits a fine tagged partition (Cousin's lemma: every gauge on a compact interval admits a fine tagged partition).

[L2]

Every nonempty set of reals bounded above has a least upper bound (Complete ordered field (least-upper-bound property)).

Verification

technique · constructive

1.1

For each $x$ , let $r (x)$ be half the supremum of the radii in $(0, 1]$ whose centered interval at $x$ lies in some member of $U$ ; openness makes this set nonempty, [L2] supplies its positive supremum, and the supremum property ensures the $r (x)$ -interval lies in at least one covering member, so $r$ is a gauge.

givenL2construct

2.1

Apply [L1] to obtain a fine tagged partition; for its finitely many tags, choose covering members containing the corresponding gauge neighborhoods, and these members cover every partition cell and hence $[a, b]$ , with a one-point interval covered by one member.

step 1.1L1choosedischarge-construct∎

Remarks

The conclusion is the interval form of the published Heine-Borel by bisection: every closed bounded interval $[a, b]$ is compact, obtained here by the distinct Cousin-lemma route.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Alessandro Fonda, The Kurzweil-Henstock Integral for Undergraduates, Ch. 1 (standard reference, not scraped)
Andrew Bruckner, Judith Bruckner and Brian Thomson, Real Analysis, Section 1.2 (standard reference, not scraped)