CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (openai/gpt-5.4)audited 2026-07-25

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Literature-sourced — the exact statement appears in a cited source; only wording and notation differ.
AI-adapted — a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated — a genuinely novel statement formulated by AI, with no source for the claim itself.

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The reverse triangle inequality

Statement

Let $F$ be an ordered field (Ordered field) and let $x, y \in F$ . Then

$\bigl| |x| - |y| \bigr| \le |x - y|.$

Facts & Assumptions

Given: An ordered field $F$ and elements $x, y \in F$ .

[L1]

Triangle inequality: $|a + b| \le |a| + |b|$ for all $a, b \in F$ (The triangle inequality).

[L2]

For every $u \in F$ , $|-u| = |u|$ , and $|w|$ equals $w$ or $-w$ (Basic properties of the absolute value).

Proof

technique · direct

1.1

Applying [L1] with $a = x - y$ and $b = y$ gives $|x| = |(x - y) + y| \le |x - y| + |y|$ , so $|x| - |y| \le |x - y|$ .

1.2

Applying [L1] with $a = y - x$ and $b = x$ , and using $|y - x| = |x - y|$ from [L2], gives $|y| \le |y - x| + |x| = |x - y| + |x|$ , so $|y| - |x| \le |x - y|$ .

L1L2

2.1

By [L2] the value $\bigl| |x| - |y| \bigr|$ equals $|x| - |y|$ or $-(|x| - |y|) = |y| - |x|$ , and each of these is $\le |x - y|$ by steps 1.1 and 1.2, so $\bigl| |x| - |y| \bigr| \le |x - y|$ .

step 1.1step 1.2L2algebra∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 9 results over 6 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

W. Rudin, Principles of Mathematical Analysis, 3rd ed., Ch. 1 (standard reference, not scraped)
T. Tao, Analysis I, 3rd ed. (standard reference, not scraped)
Purdue University analysis notes: Ordered fields and absolute value (standard reference, not scraped)