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  "body": "\n## Statement\n\n$$\\pi \u003e 3.$$\n\n## Remarks\n\nThe [[cosine series|08W5RN375X]] alternates. At $x = 3/2$, its terms decrease in size from the term $x^4/4!$ on. So $\\cos(3/2) \\ge 1 - \\frac{x^2}{2} + \\frac{x^4}{24} - \\frac{x^6}{720}$ at $x = 3/2$. This is about $0.070 \u003e 0$.\nSince $\\cos 2 \u003c 0$, the zero of $\\cos$ in $[0, 2]$ lies in $(3/2, 2)$. So $\\pi/2 \u003e 3/2$.\nArchimedes proved the sharper bounds $\\frac{223}{71} \u003c \\pi \u003c \\frac{22}{7}$. Mathlib states the claim as `Real.pi_gt_three`.\n\n## Explanations\n\n### L1 human\n\nWe show that $\\cos x$ is still positive at $x = 3/2$.\nSo its first zero $\\pi/2$ comes later, and $\\pi \u003e 3$.\n",
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