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  "body": "\n## Statement\n\nFor all real numbers $a$ and $b$,\n$$\\bigl| |a| - |b| \\bigr| \\le |a - b|.$$\n\n## Remarks\n\nProof: the [[triangle inequality|03HF5PMY5T]] gives $|a| = |(a - b) + b| \\le |a - b| + |b|$, thus $|a| - |b| \\le |a - b|$. Exchange $a$ and $b$ for the other sign.\nIt shows that the function $x \\mapsto |x|$ is continuous on $\\mathbb{R}$.\n\n## Explanations\n\n### L1 human\n\nTwo numbers that are close to each other have absolute values that are also close.\nThe difference of the distances to $0$ is at most the distance between the two numbers.\n",
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