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  "body": "\n## Example\n\nLet $n \\ge 1$ be a natural number and let $y \\ge 0$ be real.\nThen there is a unique real $x \\ge 0$ with $x^n = y$.\nFor existence, let $b = \\max(1, y)$. The function $p(x) = x^n$ is [[continuous|0C5TP6RYSV]], with $p(0) = 0 \\le y$ and $p(b) \\ge b \\ge y$.\nThe [[intermediate value theorem|00KAFBR32E]] gives $x \\in [0, b]$ with $x^n = y$.\n\n## Remarks\n\nUniqueness holds because $x \\mapsto x^n$ is strictly increasing on $[0, \\infty)$.\nThe number $x$ is the $n$-th root $\\sqrt[n]{y}$.\nFor example, it proves that $\\sqrt{2}$ exists as a real number.\n\n## Explanations\n\n### L1 human\n\nThe graph of $x^n$ starts at $0$ and grows past $y$.\nSince the graph has no gaps, it reaches the height $y$ at some point, which is the root.\n",
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