LemmaStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-01

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If every finite interval cover of $A\subseteq\mathbb{R}$ has total length at least $c$ , then every rectangle cover of $A\times[0,d]$ has total area at least $cd$

Statement

Let $A\subseteq\mathbb R$ . If every finite interval cover of $A$ has total length at least $c\ge0$ , then every finite rectangle cover of $A\times[0,d]$ , $d\ge0$ , has total area at least $cd$ .

Facts & Assumptions

Given: A finite rectangle cover and the stated interval-cover lower bound.

[L1]

Rectangles and grids are Axis-parallel rectangles in $\mathbb{R}^m$ and their volume and Grid partitions of a rectangle in $\mathbb{R}^m$ , their cells, refinements and mesh.

[L2]

Finite sums split and distribute (Finite sums and finite products, by recursion, Laws of finite sums and finite products).

Proof

technique · direct

1.1

If $d=0$ , then every covering area is nonnegative and the required lower bound is $cd=0$ . Hence assume $d>0$ . Clip the rectangles to a common bounding rectangle and partition the nondegenerate interval $[0,d]$ at every vertical endpoint.

L1given

2.1

On each nondegenerate horizontal strip, choose an interior height. The horizontal projections of the rectangles active at that height cover $A$ , so their total widths are at least $c$ .

givenstep 1.1choose

3.1

Multiply the inequality for each strip by its height and sum. Reindexing the nested finite sums counts each covering rectangle by its width times its total active height, at most its area. Thus the covering area is at least $c\sum\text{strip heights}=cd$ .

step 2.1L2algebra∎

Depends on

Used by

((0,1)∖ S)×(0,1) is bounded and open, but its boundary has positive Jordan outer content Counterexample
The Smith–Volterra–Cantor slab S×[0,1] is compact and not Jordan measurable Counterexample

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 85 results over 19 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

J. Lebl, Basic Analysis, Jordan Measurable Sets (standard reference, not scraped)
J. Lebl, Basic Analysis, Outer Measure and Null Sets (standard reference, not scraped)

If every finite interval cover of A⊆RA\subseteq\mathbb{R}A⊆R has total length at least ccc, then every rectangle cover of A×[0,d]A\times[0,d]A×[0,d] has total area at least cdcdcd